If it cools on propane, but not on electric, it shouldn't be a cooling unit problem in the sense of the chemicals not being appropriately mixed and/or blockages.
What you describe points to the electric heating element not heating or not getting hot enough. It could be an issue with the control board, the element itself, or AC power to the refrigerator. Is it gas/electric or gas/battery/electric? If it has a battery mode, does it work on that setting? The control board will use 12V whether the refrigerator has a battery mode or not since it must operate in gas mode without AC.
Assuming it doesn't have a battery mode, the first/easiest thing to check is verifying AC at the outlet in the outside compartment at the back of refrigerator.
