I recently had a simular problem in the bigger of the two A/C units on our recently traded for '10 31' Classic. Not exactly like yours, but simular.
The symtoms were the compressor unit would never shut off, and the only way you could stop it was to trip the breaker going to that A/C unit. However, the unit would cool well if the thermostat was set properly, but if the trailer ever reached the set temperature, rather than cycle the compressor, it would switch to heat mode and stay that way until the thermostat told it to cool again.
Since we were at a week-long rally and needed A/C, I only ran the unit in the heat of the day relying on the smaller rear unit at night, and then set the thermostat to such a low temp I knew it would never reach.
I called Dometic, and they told me they thought the problem was stuck relay, which I had already assumed. So, after returning home, I took the unit apart, comfirmed there was a bad relay on the board, removed it, ordered a new relay and a new start capacitor, installed them, and now the A/C unit is working properly. I would not recommend doing this board repair to anyone not experienced at soldering and board level work.
I believe this problem was caused by a bad start capacitor as I had noticed the compressor had a hard time starting in the past even though we did have good line voltage.
And no, the electrolitic start capacitors cannot be installed backwards. The relay that controls the compressor, at least on my unit, is a SPST (single pole, single throw), normally open relay, and so should work either way, however I did notice there was a land on the board that sends a power signal somewhere (I did not trace it out to find out where), so the wires should be installed as original. (I took pictures with my phone before removing the board so I could make sure I returned all the wires as they were originally).
Hope my expereince is somehow helpful to you.