Advice on Repairing Hand-Me-Down Kegerator (BM23-B)

[napalmnorm]

I was recently given a hand-me-down kegerator, a Beverage-Air BM23-B, that is probably ~10 years old. Overall the kegerator is in good shape, with some rust / buildup on components in the compressor cabinet. The previous owner said it probably hasn't been ran in over a year, and he stopped using it because it stopped getting cold, and he did not want to bother with fixing it.

I've plugged it in and confirmed the Evaporator fan turns on automatically and doesn't stop running. Turning the knob on the thermostat does kick on the condenser fan. I've read the thermostat is a common problem with these units, but since the condenser fan turned on I'm assuming the thermostat is fine. I'm assuming the only components left to check then would be the condenser & lines, the relay, and the overload. There is no charging ports, so a quick recharge is not a real option and I'm not sure if the compressor is even turning on as the vibration and noise from the condenser motor makes that difficult to tell by feel / sound. I can get a multi-meter for testing the electrical, but I am looking for advice onto how / what I should be checking with that, and how I should triage this from here.