Sorry for your troubles! I would disconnect from shore power, and turn off the 12-volt circuit breaker for the furnace and stat if there are two breakers. There may also be a 12-volt breaker for the A/C and stat control circuit?
Wait 10 seconds and flip all breakers back on, and do the Dometic hard reboot of the stat again per Dometic manual that came with your AS paperwork. Make sure the date and time are set correctly +/-. Plug back in to shore power at any time.
"Dometic stat hard reboot" threads in case there are any new ideas there:
https://www.google.com/search?q=Dome...=airforums.com
Are your batteries fully charged? On shore power what is the battery voltage on your monitor? [should be 13.7 +/-] After unplugging and waiting an hour, what is the battery voltage percentage? [should be 12.7 +/-] Any chance your battery charger/converter is failing?
Are all the other 12-volt functions working correctly?
Good luck,
Peter