Participants in this research study have ovarian cancer, fallopian tube cancer, or primary peritoneal cancer that has regrown or not responded after prior treatment. Participants will receive either paclitaxel or cabozantinib. Paclitaxel, which when administered weekly is associated with reduction in the growth of ovarian cancer in many women, and is associated with relatively mild side effects. Cabozantinib works by slowing or stopping the spread of cancer cells by blocking new blood vessel formation. New blood vessel formation is required for tumor growth. Cabozantinib is experimental in the treatment of ovarian, fallopian tube, and primary peritoneal cancer. The purpose of this study is to assess the effectiveness of the drugs paclitaxel versus cabozantinib in treating these cancers and to determine the types and severity of side effects caused by treatment with these drugs. Up to 15 people will take part in this study conducted by investigators at the University of Iowa Approximately 102 patients will take part in this study among all GOG institutions.