You said you unplug'd the molex plug in on the compressor. You mean the "overload/relay" right, not molex plug in on the compressor. And when you do that, it fires up and runs ok.
With the overload/relay "off" do you get 120 volts to the common female, if so that eliminates the board out put, and its (replace the overload/relay on the compressor) If you "don't" get 120 volts when the fridge is calling for cooling, at the common, then its the relay on the board where the 120 comes from, thus replacing the board.
I got a jumper, that I attach directly to the compressor leads, common, start, run terminals, and fire up the compressor to eliminate it as the cause. Brent@CanBC