This phase I trial tests the safety, side effects and best dose of azacitidine in combination with belinostat and how well the combination works in treating patients with follicular helper T cell lymphoma (TFH) that has come back after a period of improvement (relapsed) or that has not responded to previous treatment (refractory). This phase I trial also tests the safety, side effects and best dose of pralatrexate in combination with belinostat and how well the combination works in treating patients with relapsed or refractory peripheral T cell lymphoma (PTCL) and cutaneous T cell lymphoma (CTCL) with large cell transformation and cytotoxic phenotype. Azacitidine stops cells from making deoxyribonucleic acid (DNA) and may kill cancer cells. It is a type of antimetabolite. Pralatrexate stops cells from using folic acid to make DNA. This may help keep cancer cells from growing and may kill them. Pralatrexate is a type of antimetabolite and a type of dihydrofolate reductase inhibitor. Belinostat blocks certain enzymes needed for cell division and may kill cancer cells. It may also prevent the growth of new blood vessels that tumors need to grow and may help make cancer cells easier to kill with other anticancer drugs. It is a type of histone deacetylase inhibitor, a type of antiangiogenesis agent, and a type of chemosensitizer. Giving azacitidine in combination with belinostat may be safe, tolerable, and/or effective in treating patients with relapsed/refractory (R/R) TFH. In additional, giving pralatrexate in combination with belinostat may be safe, tolerable, and/or effective in treating patients with R/R PTCL and CTCL with large cell transformation and cytotoxic phenotype.