We have made several trips from WI to TX in the winter and never ran the furnace while traveling. Only turn it on at our stops.
There may be some risk in running your furnace while traveling as some of the heat ducts might be covered by the slides.
We normally run the heat the day before leaving as that is our packing day. Then turn the furnace off as we travel.
I have the coach water system winterized so we just jugs of water for our needs till we get to a warmer climate. Usually Mississippi.
This has always worked for us.
Peace
Dave