In auto on electric and then swaps to gas........that's due to loss of AC power
Runs on gas then 'check' light..that's due to loss of flame/won't light
Doesn't cool on DC......that could be a bad DC electric heat element
BUT ALL problems could be faulty DC power.......
Good 12V DC power is needed. Doesn't take much corrosion/bad connections to stop 12V DC.
Check for obvious bad connections on lower board.
Check ground connections clean & tight
Lower control board/module is the main power board.
Pull it and have it checked at an RV repair shop on a bench board tester