Testing Dabrafenib and Trametinib With or Without Hydroxychloroquine in Stage IIIC or IV BRAF V600E/K Melanoma

Inclusion Criteria: * Patient must have locally advanced unresectable stage IIIC or stage IV melanoma * Patient must have BRAF V600E or BRAF V600K tumor genotype based on a Clinical Laboratory Improvement Act (CLIA) approved assay * Patient must have serum LDH \> Upper limit of normal per institution standards * Patient must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Baseline measurements of sites of disease must be obtained within 3 weeks prior to study randomization * Patient must have been treated with prior immune checkpoint inhibitor therapy (anti PD-1 antibody, anti-CTLA-4 antibody or a combination regimen including either or both agents) either in the adjuvant or metastatic setting. Patient may have received investigational agents in combination with standard therapy, as long as it was adhering to the timeframes below * Patient must have discontinued active immunotherapy (IL-2, interferon, anti-CTLA-4 antibody, anti-PD-1 antibody etc.) or chemotherapy at least 4 weeks prior to randomization * Patient must have discontinued any oral targeted therapy at least 2 weeks prior to randomization * Patient may have been treated with prior adjuvant therapy including combined BRAF and MEK inhibitor therapy. Patients will be eligible if they tolerated this therapy and did not discontinue the therapy due to toxicity AND \>= 6 months have elapsed since the end of adjuvant BRAF and MEK inhibition. * Patient may have been treated with prior chemotherapy or radiation therapy * Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * 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 * All females of childbearing potential must have a blood test or urine study within 14 days prior to randomization to rule out pregnancy. * Women of childbearing potential and sexually active males must not expect to conceive or father children by using accepted and effective method(s) of contraception or abstaining from sexual intercourse for the duration of their participation in the study and for 4 months after the last dose of protocol treatment * Patient must have recovered from clinically significant reversible toxicities from previous treatment prior to randomization. Abnormal laboratory values may be grade 1, as long as they meet the eligibility criteria for organ and marrow function * Patient must be able to swallow and retain oral medication and must not have any clinically significant gastrointestinal abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels * Patient must have the ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver and/or family member available will also be considered eligible * Absolute neutrophil count \>= 1,500/mcL (obtained =\< 14 days prior to protocol randomization) * Platelets \>= 100,000/mcL (obtained =\< 14 days prior to protocol randomization) * Total bilirubin =\< institutional upper limit of normal (ULN) (obtained =\< 14 days prior to protocol randomization) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional ULN (obtained =\< 14 days prior to protocol randomization) * Creatinine =\< 1.5 x institutional ULN (obtained =\< 14 days prior to protocol randomization) * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * 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 * Patient with asymptomatic new or progressive brain metastases (active brain metastases) are eligible if the treating physician determines that CNS specific treatment is not required * NOTE: Patient with treated brain metastases are eligible. No brain imaging is required, however, 1 week must elapse after gamma knife therapy. Patient treated with whole brain radiation that have been stable for 2 months are eligible. Patient are excluded if they have leptomeningeal disease or metastases causing spinal cord compression that are symptomatic or untreated or not stable (documented by imaging) for at least 3 months or requiring corticosteroids. Patients on a stable dose of corticosteroids for at least 1 month or who have been off of corticosteroids for at least 1 week are eligible Exclusion Criteria: * Receiving any other investigational anticancer therapy during the period on study or the 4 weeks prior to randomization * Patients received BRAF and MEK inhibitor therapy in the metastatic setting * Patients who are experiencing an objective partial response to immunotherapy at the time of study enrollment * Pregnant or breast-feeding; due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used. A female of childbearing potential is defined as any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * A history of interstitial lung disease (ILD) or chronic pneumonitis * NOTE: If there is radiographic evidence of ILD that is clinically insignificant and asymptomatic, the patient would be eligible * Porphyria or psoriasis due to risk of disease exacerbation unless the disease is well controlled and they are under the care of a specialist for the disorder who agrees to monitor the patient for exacerbations * Previously documented retinal vein occlusion * A history or evidence of increased cardiovascular risk including: * Left ventricular ejection fraction (LVEF) \< institutional lower limit of normal measured within 14 days prior to randomization * A QT interval corrected for heart rate using the Bazett's formula \>= 480 msec * Current clinically significant uncontrolled arrhythmias. Exception: Patients with controlled atrial fibrillation for \> 30 days prior to randomization are eligible * Acute coronary syndromes (including myocardial infarction and unstable angina), coronary angioplasty, or stenting within 6 months prior to randomization * Abnormal cardiac valve morphology (\>= grade 2) documented by echocardiogram unless a cardiologist concludes the valve abnormality is not clinically significant. Patients with grade 1 abnormalities (i.e., mild regurgitation/stenosis) are eligible * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better * Serious concurrent infection or medical illness, including psychiatric disorders, which would jeopardize the ability of the patient to receive the treatment outlined in this protocol with reasonable safety * Receiving concurrent therapy for their tumor (i.e. chemotherapeutics or investigational agents). Radiotherapy delivered to palliate pain is allowed as long as it is not targeting a lesion that meets RECIST criteria for progression. Radiation therapy to the surgical bed with gamma knife radiotherapy while on treatment during the first cycle is allowed for small volume surgically resected brain metastases. Gamma knife radiotherapy for known active, asymptomatic small volume central nervous system (CNS) lesions may be performed during the first cycle while on study. Radiotherapy for new CNS lesions identified beyond the first cycle is not allowed on study * Immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to study drug, or excipients or to dimethyl sulfoxide (DMSO) * Prior cytochrome P450 enzyme -inducing anticonvulsant drugs (extended-interval aminoglycoside dosing \[EIADs\]) (i.e. phenytoin, carbamazepine, phenobarbital, primidone or oxcarbazepine) within 4 weeks prior to randomization * Current use of a prohibited medication described in the protocol due to potential drug-drug interaction