Since you are running the fridge on shore power, the cooling is coming from 120vac power heaters,rather than propane. On the control board you will find 5 amp glass fuses which provide the 120 vac power to the heaters.
Two things you can do: check the fuses(s) to see if they are blown or switch to propane.
If the cooling unit on the fridge is working, it doesnt care if the heat comes from the propane flame or from the 120 vac heaters.
If neither system causes the fridge to cool, there is a good chance your cooling unit is bad. However, it is also possible your control board is bad. Check the easy stuff first and then go to the more involved troubleshooting stuff. The norcold system does have built in diagnostics you can access from the front control panel. The procedures are available online.