I have a veterinary hospital and we use 15 amp rated cage dryers to dry the animals after baths. Lately we have been melting the end plugs(15 amp) on these
Dryers. Our circuit breakers are 20 amp. Should I get dedicated 20 amp sockets
Or do I then run the risk of burning the dryers themselves. Will 20 amp sockets and end plugs and a 15 amp breaker work to prevent overheating or will the draw of these older dryers just keep tripping the breaker?

Yes, use commercial or specification grade 20 am rated receptacles, be sure to use the screw type connection, and use good commercial or spec grade 20 amp or 15 amp plugs.

The issue is the heat is created at the plug connection due to looseness. Using quality grade devices should help eliminate this issue.

Always be sure each plug is pushed in tight before using the dryers. Develop an inspection routine.