To answer your question is no. After you unplug from shore power the frig should switch over to 12v automatically. If it doesn't your circuit board might be bad.
My fridge would only work on shore power too... I checked the fuse panel(under steering wheel) and there was corrosion on the fuse feeding 12v to my fridge. After wiggling it around a bit, my fridge worked off battery power for the rest of the season. This year I replaced and rewired my fuse block... No more worries.
Hiya, Im same as you, just bought the same boat as you, same year, fridge does work of the battery, how are you getting on with the sewage tank and hot water?
Shorepower on the boat is 120VAC. The fridge runs on 120VAC on shorepower. What is your shorepower? The main breaker and the fridge breaker must both be on as well as the on switch in the fridge.