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 p…