There is a capacitor (sometimes in this context confusingly called condenser) that often fails, so you might try replacing it. You would look for one with the same voltage and MFD (micro farad) specs. These things are involved on getting the motor going. Look at some YouTube video to get an idea of what these look like.

Other than that and at that age, I would replace the A/C. While RV stuff, like it is for boats, is expensive, there is some competition for A/C units and it doesn't have to be Dometic. Basically you want to match what will fit on the roof cutout hole. There will be a large rubber gasket to prevent water leaks. I think the size may be somewhat standard nowadays. Also, remember that the electrical wiring there was intended for some maximum load. So, don't go crazy thinking you can put in a much higher BTU unit. Look at the watts the old unit used and don't get something much greater.

Good luck!