Driving with your propane on is a decision that every RV’er with a gas / electric fridge has to make. There are not laws regarding it with the exception to propane is not allowed in some tunnels and you are supposed to turn off the open flame when fueling up.
We made the decision to keep our propane off during travel when we were still fairly new RV’ers. It was a recent issue with our stove falling out that brought to light something we hadnt through about before – Propane OFF…. there is still fuel in those lines.