I hope you know that I'm not trying to make you admit that what you thought was going on was wrong or anything like that. I just really don't understand these electronic things and really just want to have it verified by you as not the problem so I can eliminate it and try to figure out what it really is.
Earlier I said they were the most likely problem, because they are. They're the only thing that moves, and they are designed to cycle only so many times. Therefore they are the most likely culprit. I can't verify they are bad without being there, that's up to you. You should have been doing tons of research on how to test them and everything else you wanted to on your own.
It could be something else on the board, such as the microcontroller, which I already said. You need to troubleshoot further, and I can't help you because I'm not trying to give you a crash course in basic electronic troubleshooting. Doing what you want requires some understanding of what the different components on that pcb are, what they do, how they work together, and what to do to test them.
So, personally, I would have tested the relay, then if I verified the relay was good I would test the next thing in line which is the transistor that switches the 12v to the coil, then back to the pin on the microcontroller. But I already talked about all that stuff. Something is making the relay stay on all the time, because the compressor/fan wouldn't be on without that relay being closed.
I got no problem admitting when I'm wrong, but I never pointed you the wrong way so.... The relays could be good, but then its the associated circuitry and I'm not the one to help you find it. If you're really interested you should start watching some circuit operation and troubleshooting videos, and check out some books from the library.