Saxicola have you looked at the iSwallow app. It has all the swallow exercises the SLP would use. There are videos so you see how it is done. I just try them and the one that is the hardest is that way because something isn't working right and I do that one throughout the day for a while. I could not swallow any food for weeks a year ago and restored it almost completely.
The "shelf" is probably the throat pockets that are supposed to prevent food from going into,your airway. Food has always stuck there since halfway through radiation for me. I feel like a cud chewing cow or rather bull sometimes hacking it up over and over till it goes down. I have learned to breath then swallow.