BiCaZO: A Study Combining Two Immunotherapies (Cabozantinib and Nivolumab) to Treat Patients With Advanced Melanoma or Squamous Cell Head and Neck Cancer, an immunoMATCH Pilot Study

Inclusion Criteria: * STEP 1 - SPECIMEN SUBMISSION * Participants must have histologically confirmed melanoma that is stage III or IV, unresectable, recurrent, or metastatic non-uveal melanoma OR Participants must have histologically confirmed squamous cell carcinoma of the head and neck (HNSCC) that is either locally recurrent and non-amendable to curative therapy (e.g., radiation, surgery) or metastatic. The primary tumor location must be the oropharynx, oral cavity, hypopharynx, or larynx. Primary tumor site of nasopharynx (any histology) or unknown primary tumor are not eligible * Note: For participants with primary oropharyngeal cancer, human papillomavirus (HPV) or p16 status must be known prior to step 1 registration * Participants must have disease presentation consistent with measurable disease. Note: Current disease measurements will not be required until step 2 registration * Participants must have had documented progression during or within 12 weeks after the last dose of PD-1 checkpoint inhibition-based therapy. Participants must have been receiving checkpoint inhibition for a minimum of 6 weeks. Participants who recur during adjuvant anti-PD1 treatment or within 12 weeks of completion of adjuvant anti-PD1 treatment are eligible if they have measurable disease and are considered unresectable * Participants with known human immunodeficiency virus (HIV)-infection must be receiving anti-retroviral therapy and have an undetectable viral load test within 6 months prior to step 1 registration * Participants with evidence of chronic hepatitis B virus (HBV) infection must have undetectable HBV viral load within 28 days prior to step 1 registration * Participants with a history of hepatitis C virus (HCV) infection must have no detectable viral load within 28 days prior to step 1 registration * Participants must not have an active infection requiring systemic therapy (except HBV, HCV or HIV as mentioned above) * Participants must not have experienced myocardial infarction or thromboembolic event requiring anticoagulation within 90 days prior to step 1 registration, unless clinically stable with ongoing medical management * Participants must have recovered to baseline or =\< grade 1 Common Terminology Criteria for Adverse Events (CTCAE) version (v) 5 toxicities related to any prior treatments, unless adverse events are deemed clinically nonsignificant by the treating investigator or stable on supportive therapy * Participants must not have received more than one prior primary radiotherapy regimen, curative or adjuvant, to the mucosal surfaces of the head and neck, with the additional following criteria: * If the primary radiation is combined with chemotherapy, a minimum of 16 weeks will be required to have elapsed between the end of radiotherapy and step 1 registration. If the radiation is given alone, a minimum of 8 weeks will be required to have elapsed between the end of radiotherapy and step 1 registration * Additional palliative radiotherapy regimens are permitted but cannot have been administered to previously treated tissue (i.e., overlapping fields are excluded) with the exception of central nervous system (CNS) radiation and must be completed at least 4 weeks prior to step 1 registration * Treatment areas should be healed with no sequelae from radiation therapy (RT) that would predispose to fistula formation * Participants must not have received prior treatment with anti-VEGF therapies for any reason * Participants must be \>= 18 years of age * Participants must have a Zubrod Performance Status 0 or 1 * Participants must have adequate cardiac function. Participants with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification and must be class 2B or better to be eligible for this trial * Participants must not have any known significant organ disfunction that, in the opinion of the treating investigator, may impact suitability for receiving combination nivolumab/cabozantinib treatment * Participants must be able to take oral medication without breaking, opening, crushing, dissolving or chewing capsules * Participants must not have malabsorption syndrome * Participants must not have active autoimmune disease requiring systemic steroids (equivalent of \> 10mg of prednisone) or other immune suppression. Exceptions: * Type 1 diabetes mellitus * Endocrinopathy only requiring hormone replacement * Skin disorders (e.g., vitiligo, psoriasis, or alopecia) not requiring systemic treatment * Conditions not expected to recur in the absence of an external trigger * Participants must not have received an organ allograft * Participants must not have a history of hemoptysis (defined as \>= 1/2 tsp of bright red blood per day) or tumor bleeding within 90 days prior to step 1 registration * Participants must not have any of the following criteria due to the possibility of increased risk for tumor bleeding with cabozantinib therapy: * Prior carotid bleeding * Tumors that invade major vessels (e.g., the carotid) as shown unequivocally by imaging studies * Central (e.g., within 2 cm from the hilum) lung metastases that are cavitary as shown unequivocally by imaging studies * Any prior history of bleeding related to the current head and neck cancer * History of gross hemoptysis (bright red blood of 1/2 teaspoon or more per episode of coughing) within 3 months * Participants must not require concomitant anticoagulation with coumarin agents (e.g., warfarin), direct thrombin inhibitors (e.g., dabigatran), direct factor Xa inhibitor betrixaban, or platelet inhibitors (e.g., clopidogrel) * Participants must not require anticoagulants except for the following: * Prophylactic use of low-dose aspirin for cardio-protection (per local applicable guidelines) and low-dose low molecular weight heparins (LMWH).