Get out your voltmeter. You got some tracing to do.
The fridge does need 12 volts, whether running on gas or electric. There are two kinds of fridges in common use, one runs only on gas or 120 volt shore power, the other also will run entirely on 12 volt. That, needless to say, sucks a lot of power out of the battery, and is usually only on rigs with multiple batteries.
When you plug in the shore power, it is also supposed to make 12 volts for recharging the battery, but will also power the fridge 12 volt circuits. You need to check to see that when plugged in, you have 120 volts at the converter and the outlets. You also need to disconnect the battery and see that you still have 12 volts. It may be that the converter is bad or there are blown fuses/circuit breakers.
After you verify 120 volts and 12 volts where it's supposed to be, check the electric heater in the fridge to see that IT gets 120 volts. If so, and still no joy on electric, it means the heater is burnt out.
That the fridge throws intermittant error messages suggests you've got a bad connection somewhere.