After having both lower wisdom teeth removed, the right side of my neck, throat, and jaw are so painful I can hardly open my mouth or swallow.

See below. It is common to have post op pain that can include sore muscles and inflammation that takes some time for it to go away. That area is very close to many muscles that perform eating and swolling tasks. The area should heal but will likely take awhile. If you are concerned call your provider and ask for clarification...There is nothing wrong with that.
Not uncommon. Wisdom tooth extraction often affects many structures in the head and neck, all located in a very small space. Post operative pain can be managed with medication. Soreness can last from a day or two to more than a week. Feel better soon.
Need to be seen. You can expect to have some degree of pain following the removal of your wisdom teeth. You may also have some limitation in opening your mouth due to the swelling and soreness of the muscles of mastication. Swelling into the neck and difficulty swallowing suggest a possible acute infection and needs immediate attention by your surgeon.