Hello JennieV, I apologize for getting your thread offtrack earlier
An oily coat in cats is sometimes caused by pansteatitis, I once had a cat with this a long time ago. It's caused by a diet high in fats (usually fish catfood) and low in vitamin E. It can also sometimes cause nodules on the skin. The cat usually doesn't like to be touched or at least acts strangley when you do so. I've no idea if your cat could have this but you might ask your vet.
He also has some crazy hairballs, but its not surprising, if he is itchy, and licks himself with the fur coming off like nuts, no wonder....
Sounds like my cat Molly a few years back. She has several food allergies and and will pull out her hair along her back and get amazing hairballs if she flares up. I now feed her a raw diet which keeps everything nicely under control. A homecooked diet would probably work too, but I see that's not an option for you. Cats can be so stubborn
Hope you can figure something out!