We keep the fridge on "auto", so while we are unplugged from shore power the fridge runs on LP. During summer months it does not take long at all for the fridge temp to rise above 40 degrees F (if the fridge is turned off after cooling down), so turning the LP off while on the road is not an option (unless it's for a short period of time, an hour at most).
When we stop to fill up (diesel) we usually are at the diesel only pump.
Depending on where you check there are just as many "professionals" that say it's ok to run LP on the road as those that say "no, keep it turned off while on the road".
Bottom line, make sure all components of your LP system are maintained appropriately and on a regular basis.