To the OP, yes,run with the propane on. No needed precautions provided your fridge is in good working order. How do you know the fridge is "on"? Aside from all the lights and such you can always go outside, open the panel and feel the heat being generated in the exhaust tube surrounding the pilot light.
I have found this thread reassuring as the vast majority confirms the safety in running with the fridge on. I will only turn mine off when on a ferry. Haven't traveled through any tunnels that require it. I don't turn it off while fueling up either. The odds of a mishap are approaching zero. Fumes? Of course there are fumes at gas stations but even a spilled gallon of gas on the floor would not produce enough concentration to ignite some 25 feet away.
Over the years I have asked, pleaded actually, for anyone to produce any proof of a fire caused by a fridge running on propane while moving down the road. So far.....zero cases. I did see some photos of motor homes burning on the side of the road but upon closer look, they were fires started in the front of the rig, up where you find the engine. One sent in a story of a guy in a van conversion who was filling his gas tank that was right next to the fridge compartment. The old guy lost control of the hose and sprayed the side of the van with gas thus ingniting the whole thing. Really? That has been the best anyone could come up with.
Lets just say that someone does produce a single case of this issue. Out of millions of trips if one ended in a terrible fire, would that keep you from turning if the fridge? I mean, if you are that concerned about possible accidents you probably shouldn't drive on a highway. Lots can, and often does go wrong there.