Molly, welcome to OCF! Im sorry your father still has problems so far past finishing his treatments. Eating is one area where the doctors usually drop the ball with their patients care. I think your father would greatly benefit from seeing a dietitian or even a speech pathologist. He should have a swallowing study done to make sure he is physically able to swallow without aspirating.
With your dad being 1 1/2 years post rads, I do not think there should be swelling. After major surgeries it will take the body a full year to recover as far as any inflammation goes. I think his mouth is just overly sensitive to everything after undergoing the brutal chemo and radiation treatments. Personally, I would think its the radiation that got him more than the chemo.
Eating takes lots of practice. I made up a list of easy to eat foods. Its probably easier for your father to eat cream based products. Try peach or banana yogurt. Those are the easiest ones to eat. Look for smooth and creamy brands. If you have a Giant/Eagle grocery chain in your area, I would suggest trying to get him to eat Giant Blended peach yogurt. I have tried many kinds and this one works best for me. Try some cream soups like cream of potato, celery or cauliflower. It can take 2 full years for a recovery from what your father has been thru. I know it seems like he should be doing better but sometimes recovery is a very long road.
List of Easy to Eat Foods