This phase II trial studies nab-paclitaxel (paclitaxel albumin-stabilized nanoparticle formulation) and carboplatin followed by response-based local therapy in treating patients with stage III or IV human papillomavirus (HPV)-related oropharyngeal cancer. Drugs used in chemotherapy, such as paclitaxel albumin-stabilized nanoparticle formulation, carboplatin, hydroxyurea, fluorouracil, paclitaxel, and cisplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them spreading. Radiation therapy uses high energy x rays to kill tumor cells. Giving nab-paclitaxel and carboplatin before chemoradiation may make the tumor smaller and reduce the amount of chemotherapy and radiation therapy needed. Assigning chemotherapy and radiation therapy based on response (response-based therapy) and giving patients who are responding well lower doses of treatment may help reduce the occurrence of side effects.