Travelling in the Winter is not a problem as long as your RV has the feature of "Heated Tanks".

Your furnace will need to run while driving in temps close to freezing to keep heat around the tanks.

We start with empty Black and Grey tanks, then add a 1 gal. jug of Non-Toxic Plumbing Anti-freeze to each tank directly into the galley sink and toilet. This will allow you to use the facilities on board and it help keep the liquids in the tanks from freezing and as long as you are moving the movement should also help prevent freeze up.

I would not leave the tanks unheated for any period of time and dump regularly.

If your RV does not have heated tanks, I would definitely drain the entire system, add the plumbing antifreeze through the pump and not use the facilities. Before doing so, ensure that all fresh water has been drained from the fresh water tank, water heater tank, both holding tanks, outdoor shower etc.