For what it's worth, I had some significant swallowing problems post-Tx, and continue to have difficulty with some things, but the swallowing study was very helpful in speeding up the improvements.
I believe I would have gotten there myself, if only out of stubborn refusal to give up, but the therapist gave me a series of exercises to perform several times a day that really helped to move things along.
I would strongly encourage you to get the study done.
It will allow you to get a good look at what physically is causing the problems, and either have the appropriate procedure performed to correct it, or get the physical therapy needed to help improve.

Good Health

Chuck