This sounds so uncomfortable!
I was just reading a post about people who got relief from Myofascial release therapy ( PT where they manipulate the muscles to stretch them and break up scar tissue). Maybe this can help your situation?
You might want to ask your doctor to refer you to a Speech and Language Pathologist ( SLP) for a swallow evaluation and therapy. The swallow evaluation will show what is going on when you swallow and the therapy can assess how you are breathing when you speak to make sure you have sufficient air to support your voice.
Good luck.