Help us out with the year/make/model of RV and fridge make and model.
Your RV fridge uses 12v while in propane mode (it needs power for the circuit board), so you apparently have 12v power. From the tow vehicle or shore power, I guess. When not conencted, does anything else work, e.g. interior lights, furnace, etc.? They all use 12v power.
Most fridges automatically engage 120v (shore power) mode when available, via an AUTO switch selection on the front. If you identify your fridge, we will know what controls and features it has.