Check the amps on the compressor when it's running. Also check to make sure the compressor is not grounding.
If the compressor checks ok then I would replace the start relay.
Do I need a starter relay? A starter relay and overload kit? or is the compressor going and I just got lucky for a day? Also looking at parts online, most show a three wire hook up while this unit only has two (red and white).
I looked up your model number and it uses start relay overload kit part # 4387535
I'm thinking the two wires you seen (red & white) are the line voltage going to the start relay and overload and the compressor has three terminals like they normally do.
Hope that helps.