Testing the Combination of Two Anti-cancer Drugs, DS-8201a and AZD6738, for The Treatment of Advanced Solid Tumors Expressing the HER2 Protein or Gene, The DASH Trial

Inclusion Criteria:

* DOSE-ESCALATION PHASE: Must have histologically confirmed advanced solid tumor including but not restricted to breast cancer, gastric or gastroesophageal cancer, colon cancer, endometrial cancer, salivary gland tumors, and hepatobiliary tumors
* DOSE-EXPANSION PHASE: Must have histologically confirmed advanced/metastatic gastroesophageal cancer (cohort A) or colorectal cancer (cohort B)
* DOSE-EXPANSION PHASE: Patients must have a biopsiable lesion and provide consent for on treatment biopsy
* Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of AZD6738 in combination with DS-8201a in patients \< 18 years of age, children are excluded from this study
* Patients must have HER2-positive or HER2-expressing tumors determined by a Clinical Laboratory Improvement Act (CLIA)-certified laboratory. As a rule, for HER2 immunohistochemistry (IHC) scoring system trastuzumab for gastric cancer (TOGA) criteria used for gastric/gastroesophageal junction (GEJ) cancers will be employed (Note: in escalation phase, for breast cancer patients that are included, breast cancer criteria can be used). Specific requirement of HER2 status is outlined below:

  * HER2 expression (1-3+) by IHC locally and confirmed centrally OR
  * HER2 expression (1-3+) by IHC tested centrally OR
  * HER2 amplification based on fluorescence in situ hybridization (FISH) or next generation sequencing
* Must have received at least one line of systemic chemotherapy for either locally advanced or metastatic disease and should have either progressed on this therapy or been intolerant to this therapy
* For tumors where anti-HER2 therapy is standard of care, patients must have progressed on at least 1 line of anti-HER2 therapy if eligible. For patients where DS8201a is approved as standard of care, prior treatment with DS8201a is not allowed
* Must have unresectable, advanced/metastatic disease
* Must have at least 1 measurable lesion on CT scan per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Patient without measurable but evaluable disease are allowed for dose-escalation phase
* Must be willing and able to provide an adequate archival tumor sample available to confirm HER2 status by Central Laboratory (if local testing is used for enrollment), else must be willing and able to provide an adequate archival tumor sample for HER2 testing centrally
* Must have Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1
* Must have life expectancy of at least 3 months
* Must have left ventricular ejection fraction (LVEF) \>= 50% within 28 days before enrollment (study drug treatment) by either an echocardiogram (ECHO) or multigated acquisition (MUGA) scan
* Must have a negative pregnancy test (if female)
* Platelets \>= 100,000/mcL (within 14 days before enrollment)

  * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment
* Hemoglobin \>= 9.0 g/dL (within 14 days before enrollment)
* Absolute neutrophil count \>= 1,500/mcL (within 14 days before enrollment)

  * No administration of granulocyte colony-stimulating factor (G-CSF) is allowed within 1 week prior to screening assessment
* Creatinine clearance \> 45/mL/min (using the Cockcroft-Gault equation) (within 14 days before enrollment)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 5 x institutional upper limit of normal (ULN) (within 14 days before enrollment)
* Total bilirubin =\< 1.5 x ULN if no liver metastases or \< 3 x ULN with Gilbert's Syndrome or liver metastases at baseline (within 14 days before enrollment)
* Leukocytes \>= 3,000/mcL (within 14 days before enrollment)
* Albumin \> 2.5 g/dL (GEJ patients only) (within 14 days before enrollment)
* International normalized ratio (INR) and either partial thromboplastin time (PTT) or activated (a)PTT =\< 1.5 x ULN (within 14 days before enrollment)
* Must have adequate treatment washout period before study treatment, defined as: Major surgery (\>= 4 weeks), radiation therapy (\>= 3 weeks; in case of palliative radiation \>= 2 weeks), systemic therapy (\>= 3 weeks; in case of investigational drug use \>= 2 weeks or 5 half-lives, whichever is longer)
* Patients who are human immunodeficiency virus (HIV) positive may participate IF they meet the following eligibility requirements:

  * They must be stable on their anti-retroviral regimen, and they must be healthy from an HIV perspective
  * They must have a CD4 count of greater than 250 cells/mcL over the past 6 months on this same anti-retroviral regimen and must not have had a CD4 count \< 200 cells/mcl over the past 2 years, unless it was deemed related to THE CANCER AND/OR CHEMOTHERAPY-induced bone marrow suppression

    * For patients who have received chemotherapy in the past 6 months, a CD4 count \< 250 cells/mcl during chemotherapy is permitted as long as viral loads were undetectable during this same chemotherapy
  * They must have an undetectable viral load and a CD4 count \>= 250 cells/mcL within 7 days of enrollment
  * They must not be currently receiving prophylactic therapy for an opportunistic infection and must not have had an opportunistic infection within the past 6 months. HIV-infected patients should be monitored every 12 weeks for viral load and CD4 counts
* For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated
* Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load
* Subjects with clinically inactive brain metastases may be included. Subjects with treated brain metastases that are no longer symptomatic and who require no treatment with corticosteroids or anticonvulsants may be included in the study if they have recovered from the acute toxic effect of radiotherapy. A minimum of 2 weeks must have elapsed between the end of whole-brain radiation therapy and study treatment
* Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate central nervous system (CNS) specific treatment is not required and is unlikely to be required for at least 4 weeks (or scheduled assessment after the first cycle of treatment), and a risk-benefit analysis (discussion) by the patient and the investigator favors participation in the clinical trial
* Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial
* HER2 antibody conjugated to a topoisomerase 1 inhibitor agents as well as AZD6738 are known to be teratogenic; thus, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for at least 7 months (women of childbearing potential \[WOCBP\] only) after the last dose of study drug. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 6 months after completion of study drug administration
* Women of non-child-bearing potential defined as pre-menopausal females with a documented tubal ligation or hysterectomy; or postmenopausal defined as 12 months of spontaneous amenorrhea (in questionable cases, a blood sample with simultaneous follicle-stimulating hormone \[FSH\] \> 40 mIU/mL and estradiol \< 40 pg/mL \[\< 147 pmol/L\] is confirmatory) are eligible. Females on hormone replacement therapy (HRT) and whose menopausal status is in doubt will be required to use one of the contraception methods outlined for women of child-bearing potential if they wish to continue their HRT during the study. Otherwise, they must discontinue HRT to allow confirmation of post-menopausal status prior to study enrollment. For most forms of HRT, at least 2-4 weeks will elapse between the cessation of therapy and the blood draw; this interval depends on the type and dosage of HRT. Following confirmation of their post-menopausal status, they can resume use of HRT during the study without use of a contraceptive method
* Male subjects must not freeze or donate sperm starting at screening and throughout the study period, and at least 6 months after the final study drug administration. Preservation of sperm should be considered prior to enrolment in this study
* Female subjects must not donate, or retrieve for their own use, ova from the time of screening and throughout the study treatment period, and for at least 7 months after the final study drug administration
* Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible

Exclusion Criteria:

* Patients with a history of (non-infectious) interstitial lung disease (ILD)/pneumonitis that required steroids, have current ILD/pneumonitis, or where suspected ILD/pneumonitis cannot be ruled out by imaging at screening. Patient using e-cigarettes/vaping are also excluded
* Patients with a medical history of myocardial infarction within 6 months before enrollment (study treatment), symptomatic congestive heart failure (New York Heart Association Class II to IV, corrected QT interval (Fridericia's formula-corrected QT interval \[QTcF\]) prolongation to \> 470 ms (females) or \> 450 ms (males) as corrected by Framingham's formula
* Patients with spinal cord compression or clinically active central nervous system metastases, defined as untreated and symptomatic, or requiring therapy with corticosteroids or anticonvulsants to control associated symptoms
* Patients with multiple primary malignancies within 2 years, except adequately resected non-melanoma skin cancer, curatively treated in situ disease, or other curatively treated solid tumors
* Patients with a history of severe hypersensitivity reactions to either the drug substances or inactive ingredients in the drug product
* Patients with an uncontrolled infection requiring IV antibiotics, antivirals, or antifungals
* Patients with substance abuse or any other medical conditions that would increase the safety risk to the subject or interfere with participation of the subject or evaluation of the clinical study in the opinion of the investigator
* Patients with a concomitant medical condition that would increase the risk of toxicity in the opinion of the investigator
* Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities grade \>1) with the exception of alopecia. Subjects with chronic grade 2 toxicities may be eligible per discretion of the investigator after discussion with study principal investigator (PI) (e.g., grade 2 chemo-induced neuropathy).
* Any previous treatment with an ATR inhibitor
* Patients with any clinically apparent pulmonary compromise resulting from intercurrent pulmonary illnesses including, but not limited to, any underlying pulmonary disorder (i.e., pulmonary emboli within three months of the study enrollment, severe asthma, severe chronic obstructive pulmonary disease \[COPD\], restrictive lung disease, pleural effusion, etc.), and any autoimmune, connective tissue or inflammatory disorders with potential pulmonary involvement (i.e., Rheumatoid arthritis, Sjogren's, sarcoidosis, etc.), or prior pneumonectomy
* Patients with myelodysplastic syndrome/acute myeloid leukemia or with features suggestive of myelodysplastic syndrome (MDS)/acute myeloid leukemia (AML)
* Patients unable to swallow orally administered medication and patients with gastrointestinal disorders likely to interfere with absorption of the study medication
* Concomitant use of known strong CYP3A inhibitors (e.g., itraconazole, telithromycin, clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir, saquinavir, nelfinavir, boceprevir, telaprevir). The required washout period prior to starting study treatment is 2 weeks. Concomitant use of known strong (e.g., phenobarbital, enzalutamide, phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine and St John's Wort ). The required washout period prior to starting study treatment is 5 weeks for enzalutamide or phenobarbital and 3 weeks for other agents
* Patients with a pleural effusion, ascites, or pericardial effusion that requires drainage, peritoneal shunt, or cell-free and concentrated ascites reinfusion therapy (CART). (Drainage and CART are not allowed within 2 weeks prior to screening assessment)
* Patients with previous allogeneic bone marrow transplant or double umbilical cord blood transplantation (dUCBT)
* Whole blood transfusions in the last 120 days prior to entry to the study (packed red blood cells and platelet transfusions are acceptable within the last 28 days as long as they are not within 1 week prior to screening assessment)
* Patients at risk of brain perfusion problems, e.g., medical history of carotid stenosis or pre-syncopal or syncopal episodes, history of transient ischemic attacks (TIAs)
* Uncontrolled hypertension (grade 2 or above) requiring clinical intervention
* Patients with relative hypotension (\< 90/60 mm Hg) or clinically relevant orthostatic hypotension, including a fall in blood pressure of \> 20 mm Hg
* Patients who have received corticosteroids (at a dose \> 10 mg prednisone/day or equivalent) for any reason within 2 weeks prior to first dose
* Patients with uncontrolled intercurrent illness
* Patients with psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study because DS-8201a is a HER2 antibody conjugated to a topoisomerase 1 inhibitor agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with DS-8201a, breastfeeding should be discontinued if the mother is treated with DS-8201a. These potential risks may also apply to AZD6738
* Patients cannot be receiving chloroquine or hydroxychloroquine. Patients receiving these drugs must have a washout period of \> 14 days before enrollment/randomization