Interesting. As you say, the pilot doesn't stop so the control card has to be getting 12 volts. It has to be getting enough gas to run the pilot but maybe it kinked enough that it cant supply the higher volume needed. Can you watch under the slide to see where the hoses fold and perhaps move the hose enough that the bends are not sharp and see if the frig and stove are then ok. Light a burner and watch it as you bring the slide in and see right where the flame dies then check the hose for kinks and remove them by tying them temporarily and try to relight.
When we had a propane fridge we never shut it off.