Are there exercises I can do to improve my swallowing?

Speech therapist. I am not an expert in this. But there are experts in this field, some doctors like ent's, and gastroenterologists who can help you diagnose the problem. But speech and swallowing therapists actually teach you what to do. Talk with your pcp first. A simple swallowing study may result in quick diagnosis and referral to the therapist. If it is more complex your md can refer you to a specialist.
See doctor. If you are having trouble swallowing, you should see your doctor to find out why.