Whether you use 120V or propane, all that either does is supply heat for the boiler in an RV refrigerator. The control circuit board is operated by 12V-dc power. If you have an ice maker, that would have 120V-ac power to the motor and the water fill valve, but no part of the refrigerator other than the heating element.
It sounds to me as though the two most likely possibilities are the thermistor that senses temperature in the refrigerator, or the circuit board.
If you can tell us what model refrigerator you have, we can probably look up to see what the value of the thermistor should be and also give you a resistance value to connect in it's place to see if it turns the heat supply off, as it should. Doing that would indicate if the problem is the thermistor or the circuit board.