This randomized phase II/III trial studies the side effects and best dose of nivolumab and ipilimumab when given together with or without sargramostim and to see how well they work in treating patients with stage III-IV melanoma that cannot be removed by surgery. Monoclonal antibodies, such as ipilimumab and nivolumab, may kill tumor cells by blocking blood flow to the tumor, by stimulating white blood cells to kill the tumor cells, or by attacking specific tumor cells and stop them from growing or kill them. Colony-stimulating factors, such as sargramostim, may increase the production of white blood cells. It is not yet known whether nivolumab and ipilimumab are more effective with or without sargramostim in treating patients with melanoma. Eligibility includes being at least 18 years old and must have known v-raf murine sarcoma viral oncogene homolog B1 (BRAF) mutational status of tumor; wild-type (WT) or mutated, prior to randomization. Contact Mercy clinical trials for further eligibility criteria.