First thing I would do is physical inspection of the AC pan for oil. Go on roof, pull the AC cover/shroud, look for oil, if it's wet - you have a leak / have lost freon. Replace the AC unit. (Dometic penguin I/II are non-serviceable for leaks) ($1k+ installed)
The next thing to do is to put an ammeter on the common wire to the compressor. Should see surge on startup and then build up to and rest at ~11amps running. If you are only seeing 6-7-8 amps then you have lost freon pressure. (I have been here - no leaking fluid detectable but amp draw on compressor motor was low / meaning less resistance than normal). Replace AC unit. ($1k+ installed)
If you are getting 0 amps then and it is simply failing to even start rotation - move to start capacitor. To check the start capacitor, first discharge it, then using your multi meter to test the start capacitor for microfarads and make sure your reading aligns with it's rating (should be printed on the capacitor - look for MFD rating). If out of range after 4-5 tests, replace the start capacitor (~$20 bucks or so).
If compressor is starting but then fails to continue to run, check the run capacitor using the same method above. Change the run capacitor if needed. (~$20 bucks or so).
If capacitors are good, but zero amps to the compressor, then check all AC connections in the system. find the loose connection. $0 fix
It is possible you have a bad board, but this is less likely. (need to check cost on Dometic Penguin board)
Bottom - line - you are looking to determine if power getting to the compressor, does it start at all / see any power, if it runs does it run at expected amps, or does it run and then stall........
That's about all I got
my guess is bad capacitor or freon leak.