Is my compressor shot?

I think that you have a motor problem, specifically with the starting switch or starting capacitor. The fact that the pump can build pressure to the switch cutout once it's running tells me the pump is okay. As the motor can drive the compressor to this point, once it is running, indicates the running mode of the motor is okay. I'm not sure how much electrical knowledge you have but there are numerous YouTube videos on how to clean the starting switch contacts and test the capacitors on a capacitor start/capacitor run motor like you have. There's no problem working on electrical equipment as long as you follow the procedures and do not do anything stupid.