Isolated Limb Perfusion With Melphalan in Treating Patients With Stage IIIB-IV Melanoma or Sarcoma

Inclusion Criteria: * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * The patient must have histological-proven primary or recurrent extremity melanoma (or other skin cancer including but not limited to Merkel cell carcinoma) or sarcoma, stage IIIB, IIIC, or IV (American Joint Committee on Cancer \[AJCC\] staging must be documented in patient's medical record, as determined by computed tomography \[CT\] of the chest, abdomen and pelvis, and/or whole body positron emission tomography \[PET\] scan, within six weeks prior to administration of study drug) * Patients with stage IV disease who have high tumor burden and extensive symptomatic extremity disease * Patients with stage IIIC disease must either have had regional lymph nodes previously removed or will have them removed either at the time of regional treatment or soon thereafter * Disease to be treated by ILP must be unresectable and distal to the planned site of tourniquet placement (which for the leg is generally the apex of the femoral triangle, or for the arm is distal to the deltoid insertion) * Patient's disease must be bi-dimensionally measurable by caliper or radiological method as defined in the Response Evaluation Criteria in Solid Tumors (RECIST) criteria (RECIST updated version 1.1); lesional tissue not necessary for diagnostic or other clinical purposes may be designated prospectively for research tissue banking * Hemoglobin \>= 8.0 g/dl * White blood count (WBC) of \>= 2000 m\^3 * Absol…