Taladegib, Paclitaxel, Carboplatin, and Radiation Therapy in Treating Patients With Localized Esophageal or Gastroesophageal Junction Cancer

Inclusion Criteria:

* Histologically or cytologically confirmed adenocarcinoma of the esophagus or gastroesophageal junction (EAC)
* Localized EAC and its baseline clinical stage determined as: T2-T3N0 or T1-3N positive (+); imaging studies suspicious for metastases must be followed with a negative biopsy before a patient can enter the study
* Patients with malignant celiac nodes are eligible if the primary lesion is in the mid-thoracic or distal thoracic esophagus or it is involving the gastroesophageal junction
* Tumor must have labeling index of \>= 5% of the nuclear Gli-1 (integral biomarker) performed in the MD Anderson Cancer Center Clinical Laboratory Improvement Amendment (CLIA) laboratory for patient to be eligible in this trial (if enough archival tissue is not available to determine labeling index, patient must agree to a biopsy to be eligible for the study)
* Tumor may not extend \> 4 cm below the gastroesophageal junction
* Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1
* All patients must be willing to provide research tumor tissue for biomarker studies at baseline, from archival tumor tissue or through endoscopy if sufficient archival tissue is not available; all patients must also allow biomarker studies on the tissue obtained through surgery to remove the primary cancer
* Phase II only: patients volunteering for the Phase II part of the protocol must be willing to undergo a research endoscopy for tissue collection on day 8 (+/- 2 days) from the beginning of therapy
* Absolute neutrophil count \>= 1500/mm\^3
* Platelets greater \>= 100,000/mm\^3
* Hemoglobin \>= 8 g/dL
* Serum creatinine =\< 2 x upper limit of normal (ULN)
* Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN
* Serum bilirubin =\< 1.5 x ULN
* Patient must be able to comprehend the approved consent document and have the willingness to sign it; the patient prior to enrollment and the administration of any protocol-specific therapy must sign the consent document
* Willingness and ability to comply with study procedures and follow-up examinations
* Must be considered medically fit for operation as determined by multidisciplinary evaluation
* Males and females with reproductive potential must agree to use 2 forms of medically approved contraceptive precautions and for at least 6 months following the last dose of biochemoradiation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
* Women of childbearing potential are defined as follows: having regular menstrual cycles; has amenorrhea, irregular menstrual cycles or using a contraceptive method that precludes withdrawal bleeding; have had a tubal ligation; women are considered not to be of childbearing potential for the following reasons: had hysterectomy and/or bilateral oophorectomy; post-menopausal defined by amenorrhea for at least 1 year in a woman \> 45 years old
* Females with childbearing potential must have a negative serum pregnancy test within 14 days prior to treatment start

Exclusion Criteria:

* Baseline clinical stage of T1N0 or inoperable T4 (unequivocal organ involvement) are to be excluded
* Unequivocal metastatic tumor at baseline
* Tracheo-esophageal (TE) fistula or direct invasion into the tracheo-bronchial mucosa; a bronchoscopy (biopsy and cytology should be performed) is required to exclude TE fistula or tracheo-bronchial involvement in patients with a tumor located at \< 26 cm from the incisors
* Cervical esophageal cancer will not be entered in this study
* Any prior chemotherapy, surgery, or radiotherapy for EAC
* Prior mediastinal irradiation (for any reason)
* Clinically significant ulcerative colitis, inflammatory bowel disease, or partial or complete small bowel obstruction are to be excluded
* Malabsorption syndrome or other condition that would interfere with intestinal absorption are excluded
* Pregnant or nursing females are to be excluded; breastfeeding should be discontinued if the mother is treated with taladegib
* Presence of other significant cancer(s) or history of other significant cancer(s) within the last 3 years (patients who have been cancer-free for 3 years, or have a history of completely resected non-melanoma skin cancer or successfully treated in situ carcinoma of the cervix are eligible)
* Known active viral or other chronic types hepatitides (hepatitis B, C) or cirrhosis
* Uncontrolled concurrent illness including, but not limited to: serious uncontrolled infection, symptomatic congestive heart failure (CHF), unstable angina pectoris, cardiac arrhythmia that interfere with blood pressure, uncontrolled diabetes, or psychiatric illness/social situations that would limit compliance with the study requirements
* Patients with uncontrolled hypocalcemia, hypomagnesemia, hyponatremia or hypokalemia defined as less than the lower limit of normal for the institution, despite adequate electrolyte supplementation
* Patients who are receiving concurrent non-protocol anti-cancer therapy (chemotherapy, radiation therapy, surgery, immunotherapy, hormonal therapy, targeted therapy, biologic therapy, or tumor embolization) are to be excluded
* Patients may not be receiving any other investigational agents
* Patients with known hypersensitivity to taxanes or platinums are to be excluded
* Patients taking medications with narrow therapeutic indices that are metabolized by cytochrome P450 (CYP450), including warfarin sodium (Coumadin) are ineligible; patients on strong cytochrome P450, family 3, subfamily A (CYP3A) inhibitors will also be excluded
* Known human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy
* Any other conditions or circumstances that would, in the opinion of the investigator, make the patient unsuitable for participation in the study