Dometic fridge

It would help if you told us the model number of your Fridge but nonetheless there is a diagram on that black fuse cover that should indicate how to remove that cover with a screwdriver. Mine has a 5 amp fuse, it's very easy to check. If it is not bad than what worked for me one time was to remove each electrical connection and clean for any corrosion (shut your power down before doing this) but I eventually had to replace the control board, not a big deal.
There are some video's on youtube to help you with some diagnosis and repair.

Sound like 12V DC is OK as it is swapping to propane when in auto and NO 120V AC detected.

SO...........
120V AC at outlet in outside compartment where power cord plugs in?
YES......
Then you need to remove cover on control module. Unplug power cord from module, unplug DC leads from module. Remove the cover screws and any clips (can be 1,2,3 Screws)

5A glass fuse is for AC ....check heater element. (44 ohms)

check all GFI's to see if any tripped

Do you have 12 V DC on allowing power to the control board?

Do you have 120VAC to the receptacle the refer is plugged into?