I am with SidecarFlip and pianotuna; the only time the valve on the main propane tank gets turned off is when refilling the propane. I'd think you'd want the propane and the fridge on if you're driving for any significant length of time, lest the ice cream become soup and the milk become sour.
The only ways I can think of that air would get into the lines with the main valve turned off, at least at any significant rate, is if one of the appliances is actually on and trying to light or if there's a leak somewhere. It might not be a bad idea to go over the various connections with bubble fluid or soapy water (with the gas on, of course). I suppose it may also be possible that the regulator is doing some strange venting things or sticking or something; they can wear out over time.