Sounds like your converter/charger (or maybe it's an inverter/charger?) crapped out and the batteries went dead. With no 12v, the fridge will stop working, even though it is using 120v for its main heat source.
Do you know where your charging system is? It won't be far from the batteries - just follow the big cables. What make & model is it? That will help other s suggest remedies.
You must have some gauge that gives battery status - what does it say? I'm guessing "dead" or "zero percent charge" or something similar. If it reads in volts, it might say 6 or 8. That's your clue there is no charging. You can also use a voltmeter to check the battery voltage directly - just hook the leads across the + and - battery terminals where the big cables attach.
2004 American Tradition; 2013 Buick Verano
Homebase in the Ocala Nat'l Forest near Ocala, FL