The refrigerator will not continue to work on propane after unplugging from shore power until you figure out the 12v problem. It has a control board that needs 12v for it to work.
As Scott mentioned, when hooked to shore power, the converter supplies all the necessary 12v to every aspect of your unit. Once unplugged from shore power, the battery takes over.
As mentioned, first check to be sure the battery is good and fully charged. Check the connections at the battery, the cut-off switch, any in-line 12v circuit breakers, ground cables and the like.
Keep us posted as to what you find.
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