You have gotten a lot of great advice but I thought I’d add a little more. I mainly ride western but I used to ride primarily English. I had a lot of problems keeping my heels down and in the stirrups. My trainer told me to try to raise my opposite arm (in your case your right arm) to try to balance myself out. I know it sounds weird, but it really works. Obviously you can’t ride down the show arena with your arm up but it helps get that muscle memory working. When you try this make sure you’re either in an enclosed space or you are on a lunge line.
Also, stirrup length (as mentioned above) is a great idea to check. Riding with no stirrups helps with your balance which will help in the show ring in case you do loose a stirrup.
Best of luck to you and please let us know your progress!