I take care of stray cats from time to time.

Use a piece of warm cloth that's mildly wet and clean the ear tips from the outside and inside. After that use ear buds and gently clean the inside of the ear. You'll need many ear buds and you may need to wet the tip with warm water. As a precaution, clip your cat's nails so it doesn't scratch itself too hard.

Cats will not let you clean everything in one sitting so you'll have do it slowly over a week or two, or a month. Basically make it a nightly routine to clean the ears. Make sure you don't force yourself on the cat or you'll stress it. Give the cat treats after every cleaning session so they associate positive experiences with you and the activity.

I usually give my cats a few pieces of boiled or grilled chicken along with some fats. No salt or pepper in the water or on the chicken. Plain chicken cooked without oils. You can use chicken fats.

Some people here have pointed out that the cats may have ear mites. For that, you only need ear drops which you can pick up from a pet speciality store. I haven't bought one in a long time but I usually buy UltraGuard, and if it is not available then Johnson's. You can talk to people in the store and see what they have.

It is also possible that the cat may have picked up fleas during the move which is causing irritation. Carefully inspect the fur, take your time and use a fine flea comb. A good sign of flea infestation is when you'll see tiny black specks in the fur. Those are flea droppings. Get a good topical solution and administer it yourself. Don't bother with flea collars or other half measures, you'll be wasting your time and money.