I don't want to get this thread off topic but after doing research on this for years there is no law on the books in any province or state that requires that propane in an RV needs to be turned off - a lot of assumptions and some scare tactics used on web sites - but no laws. Many manufacturers advise the use of propane when traveling so this would be going against the law if it were so. The only laws on the books are in effect in some jurisdictions with the operation of propane in tunnels (this is being revised or reviewed in a lot of areas with more and more propane & NG vehicles coming into operation as well as better venting of tunnels), propane on ferries (not allowed and will be checked and tanks tagged in the off position), Parking garages (most RV's do not fit so not an issue) and in refueling (this is listed at the pumps when you are getting fuel but to be truthful there is not one documented case of a propane appliance causing a fire at a gas station (again a lot of "I heard ....." but nothing that was ever proven to be from an RV appliance that I have seen)).
Propane in an RV is a completely different set of rules than propane in the back of a car or truck (many of the listings on Google quote carrying propane tanks in a car or truck as being the same rules as an RV - not so). A propane tank attached to a barbeque in the back of a truck must be shut off as this is not a permanent or a registered use for such an appliance.
Likewise supposed comments of "this accident was made so much worse because the propane tanks were open" are again false. I have been to a number of accidents that involved RV's and it is just part of the procedure to check the tanks and if on shut off. Just like disconnecting the battery and stopping fuel leaks. I have never seen or heard an actual report about propane making an accident scene worse.
As far as common sense I always hate when this is tossed out as it is a general assumption that what one person thinks everyone should follow and if you don't follow it then you must be stupid. Common sense to some is never leaving the house as accidents happen, to others it means something completely different.
With all the above I am not saying that there has never been a problem with propane. I am sure that there are problems, have been problems and will be problems in the future. But if used correctly there is nothing more to fear from propane than any other item in the RV - and this includes using it when in motion for keeping the fridge cold.