I agree with Barry you need to get it checked, to answer your question "Will the fridge continue to run off the battery power?" the answer is yes Battery + Propane will run it but not battery alone without propane. Shore power (or Generator) will run it without propane and/or battery (it always needs 12 volt for the control circuit but if no battery connected the TT converter will provide the 12 volts).
Dale & Terri, Lulu & Tiki (our Chihuahua's), New rescue puppy Prince - Pom/Pug mix.
2013 Tundra, Double Cab, 5.7, TRD Off-Road, 4X4, Full Tow Package
2013 Wildwood T26TBSS - Sold
2000 Gulf Stream Sun Voyager M-8357 MH, Ford Trident V10 Gasser, 35 foot.