Nab-paclitaxel and Carboplatin Followed by Response-Based Local Therapy in Treating Patients With Stage III or IV HPV-Related Oropharyngeal Cancer

Inclusion Criteria:

* Patients must have pathologically confirmed HPV-positive squamous cell carcinoma
* HPV testing must follow the following criteria

  * HPV testing using an E6/E7 based assay is preferred, and does not require any validation (e.g. HPV in situ hybridization \[ISH\] or HPV E6/E7 polymerase chain reaction \[PCR\])
  * For oropharyngeal tumors p16 immunohistochemistry (IHC) positivity is sufficient to enroll and initiate treatment (p16 IHC interpretation to follow guidelines by Jordan/Lingen et al 2012); it is recommended that p16 IHC positivity is validated at a later point (during or after treatment) using an E6/E7 based test at the University of Chicago and provided slides will be used
  * For non-operative (OP) tumors accurate HPV testing (i.e. ISH, or E6/E7 based testing) is required for enrollment and treatment initiation
* Availability of \>= 10 unstained 5 micron slides
* Patients with American Joint Committee on Cancer (AJCC) (7th edition, 2010) nodal stage N2 or N3 or a T4 primary tumor
* The primary and nodal involvement must be assessable on clinical exam (mucosal and lymph node exam)
* The primary and nodal involvement must have been defined bi- or uni-dimensional measurements measurable by RECIST
* No previous radiation or chemotherapy for a head and neck cancer
* No surgical resection for a head and neck cancer within 8 weeks of enrollment (although lymph node biopsy including excision of an individual node with presence of residual nodal disease, or surgical biopsy of the tumor is acceptable)
* Eastern Cooperative Oncology Group (ECOG) performance status 0-1 (Karnofsky \>= 70%)
* Leukocytes \>= 3000/mm\^3
* Platelets \>= 100,000/mm\^3
* Absolute neutrophil count \>= 1,500
* Hemoglobin \> 9.0 gm/dL
* Albumin \> 2.9 gm/dL
* Total bilirubin =\< 1.5 mg/dl
* Creatinine clearance \> 45 mL/min (or serum creatinine \[SCr\] =\< 1.5 mg/dL), normal within 2 weeks prior to start of treatment
* The standard Cockcroft and Gault formula or the measured glomerular filtration rate must be used to calculate creatinine clearance (CrCl) for enrollment or dosing
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 X upper limit of normal (ULN)
* Alkaline phosphatase =\< 2.5 X ULN
* Patients must sign a study-specific informed consent form prior to study entry; patients should have the ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Unequivocal demonstration of distant metastases (M1 disease)
* Intercurrent medical illnesses which would impair patient tolerance to therapy or limit survival; including but not limited to ongoing or active infection, immunodeficiency, symptomatic congestive heart failure, pulmonary dysfunction, cardiomyopathy, unstable angina pectoris, cardiac arrhythmia or psychiatric illness/social situations that would limit compliance
* Pregnant and nursing women are excluded; men and women of child-bearing potential are eligible but must consent to using effective contraception during therapy and for at least 3 months after completing therapy; women with child-bearing potential must have a negative serum or urine beta-human chorionic gonadotropin (B-hCG) pregnancy test at screening
* Other coexisting malignancies or malignancies diagnosed within the previous 3 years no evidence of disease for at least 3 years; exceptions to this include non-melanoma skin cancer, cervical cancer in situ, well differentiated thyroid cancer or prostate cancer; other cancers that per assessment of the PI are not prognosis limiting can be allowed after review by the PI
* Prior surgical therapy other than incisional or excisional biopsy and organ-sparing procedures such as debulking of airway-compromising tumors or neck dissection in a patient with an unknown primary tumor; residual tumor is required for enrollment on study
* Patients receiving other investigational agents
* Peripheral neuropathy \>= grade 1