CBL0137 in Treating Patients With Advanced Extremity Melanoma or Sarcoma

Inclusion Criteria: * Patient must have a life expectancy of \> 6 months. * Have an Eastern Cooperative Oncology Group (ECOG) Performance Status of 0-2. * Patients either: * Must not have undergone any limb-directed treatment OR * Have undergone a previous Melphalan based regional therapy for which they did not have a complete response and, present with persistent, progressive, or recurrent disease. * \* NOTE: Patients with indeterminate staging must be reviewed by the Principal Investigator prior to registration. * Patient must have had a washout period for at least 30 days or 5 half-lives from any prior chemotherapy, radioactive, or hormonal cancer therapy, or 4 weeks from any checkpoint inhibitors or other biologic (including TVEC), whichever is longer * Patient must have histologically proven primary or recurrent extremity melanoma (stage IIIB, IIIC, or IV), or advanced extremity sarcoma not amenable to surgical resection * (American Joint Committee on Cancer \[AJCC\] melanoma staging must be documented in patient's medical record, as determined by computed tomography \[CT\] of the chest, abdomen and pelvis, within six weeks prior to administration of study drug; * Due to the heterogeneous nature of sarcoma, AJCC sarcoma staging is NOT required * Patients with Stage IIIC disease must either have had regional lymph nodes previously removed or have stable or regressed disease on imaging from prior systemic therapy (defined as modified RECIST 1.1 SD, CR, or PR). * Stable or regressed disease must be present for at least the 2 months prior to IA CBL0137 and patient is no longer receiving systemic therapy (with the exception of immunotherapy) during this time period for melanoma. * Stable or regressed disease must be present for at least the 2 months prior to IA CBL0137 and patient is no longer receiving systemic therapy during this time period for sarcoma * Patients with Stage IV disease must have had all distant disease resected at least 30 days prior to regional treatment, or exhibit stable or regressed disease .on imaging from prior systemic therapy (defined as modified RECIST 1.1 SD, CR, or PR). * Stable or regressed disease must be present for at least the 2 months prior to IA CBL0137 and patient is not receiving systemic therapy (with the exception of immunotherapy) during this time period for melanoma * Stable or regressed disease must be present for at least the 2 months prior to IA CBL0137 and patient is no longer receiving systemic therapy during this time period for sarcoma * Melanoma or sarcoma patients who have stable or completely responded brain metastases from previous gamma knife surgery and/or systemic therapies are eligible. * Patient's disease must be measurable by caliper or radiological method as defined in the modified Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria. * Patient must have adequate bone marrow, liver and renal function as assessed by the following: * Hemoglobin \>= 9 g/dL. * White blood count (WBC) of \>= 3000 m\^3. * Absolute neutrophil count (ANC) \>= 1,500/mm\^3. * Platelet count \>= 100,000/mm\^3. * Total bilirubin =\< 1.5 x upper limit of normal (ULN). * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x the ULN. * Creatinine clearance (CrCl) \> 45 mL/minute. * Patient must have a palpable femoral/radial pulse in the affected extremity. * Patients must have recovered from adverse events from previously administered agents (\<=grade 2) prior to first study drug administration * Participants of child-bearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Ability to read and understand English and the ability to complete paper and/or electronic survey assessments. * Participant must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure. Exclusion Criteria: * Cardiac disease: Congestive heart failure \> Class II New York Heart Association (NYHA). Patients must not have unstable angina (angina symptoms at rest) or new onset angina (began within the last 3 months) or myocardial infarction within the past 6 months * Males with mean QTcF values of \>450 msec and females with QTcF values of \>470 msec, patients who are known to have congenital prolonged QT syndromes, or patients who are on medications known to cause prolonged QT intervals on ECG. * Use of drugs known to prolong QT. * Patients with known hypersensitivity to any of the components of CBL0137. * Uncontrolled hypertension defined as systolic blood pressure \> 150 mmHg or diastolic pressure \> 90 mmHg, despite optimal medical management. * Thrombotic ((excluding prior catheter-related thrombus that has been adequately treated) or embolic events such as a cerebrovascular accident including transient ischemic attacks within the past 6 months. * Patients with symptoms or signs of vascular insufficiency. Specifically, patients with any history of blood clots (excluding prior catheter-related thrombus that has been adequately treated) or lifestyle altering ischemic peripheral vascular disease will be excluded. * Evidence or history of bleeding diathesis or coagulopathy. * Patients with known heparin induced thrombocytopenia. * Untreated or growing brain metastasis: Patients with neurological symptoms must undergo a CT scan/magnetic resonance imaging (MRI) of the brain to exclude untreated or growing brain metastasis. * Known human immunodeficiency virus (HIV) infection or active hepatitis B or C. * Active clinically serious infection \> Common Terminology Criteria for Adverse Events (CTCAE) Grade 2. * Serious non-healing wound, ulcer, or bone fracture. * Major surgery or significant traumatic injury within 30 days of planned intra-arterial infusion. * Current treatment or, treatment within the previous 24 months, for another non-melanoma or sarcoma malignancy. * Patients who have already received 2 prior infusions of CBL0137. * Pregnant or nursing female participants. * Psychiatric conditions or diminished capacity that could compromise the giving of informed consent, or interfere with study compliance. * Unwilling or unable to follow protocol requirements. * Any condition which in the Investigator?s opinion deems the participant an unsuitable candidate to receive study drug. * Received an investigational agent within 30 days prior to enrollment.