I have not pulled the thermostat on that specific refrigerator, but as I recall mostly you need access to everything, so the refrigerator will probably need to be removed. Then you can get at the thermostat, the capillary tube that goes into the box (and needs to be carefully handled on the new 'stat so you don't kink it). Then you can see all the components and figure out how to replace it.
Now, my solution would be just to forget it, and use the refrigerator on propane only. The cost of a new thermostat, the hassle of replacement and so on don't justify the effort. But then, I virtually never use the 120 volt side of any of my RV refrigerators. I don't have to change over, light them, fiddle with them from one system to the other. I am on a trip right now, and at a friends home, plugged into their power. But the refrigerator is on propane, as it was when I left.