I would look at whether it is trying to cool or not. Fire up the propane and look at the flame. While it is on, the unit is trying to cool and once off it thinks it is down to temp. It is easier to see if the fridge is trying to cool while on propane versus 120 volts.
If trying to cool but no cooling is taking place, you may have a major problem. If the flame shuts off before it should, it could be a thermostat or possibly a control board.
These fridges heat up ammonia in the tubing that does the cooling as it flows through. A clog or leak will ruin your day. As these units operate on propane and 120 volts, there are 2 totally separate functions for heating the ammonia. If both are not working you look for what is common to both.
Retired electronic technician (45 years in the field)
2017 Greyhawk 29W
solar & many other mods
wife (maybe I should have given her top billing)