Does fridge run on propane when connected to shore power?
As posted above.......
Fridge needs good 12V DC power to run on electric heater element or on propane.
If fridge operates on electric and propane when connected to shore power then YES new batteries will fix your issue of fridge not running on propane when NOT on shore power.
But if fridge doesn't run on propane when on shore power........then you have other problems then just bad battery
You really need to get a new battery......it is what supplies DC power to trailer brakes should trailer become disconnected from vehicle when towing.
By LAW it is required.