My youngest sister went through the same thing after her last RAI. Her salivary glands actually became infected and had to be surgically drained. Talk to your endo. There are a lot of things that can lead to dry mouth and you could be showing a symptom of another condition. It's a common complication of RAI but shouldn't be going on this long.
Here's something I found on line that may help:
Other medical conditions can be causes of dry mouth as well including Alzheimer’s disease, endocrine disorders, stroke, anxiety, depression, and various autoimmune diseases. If dry mouth comes on slowly, it is most likely a result of aging. If it occurs suddenly, mention the condition to your doctor.