I have a similar situation every year. I travel from the low warmer desert country to a high elevation into potential snow and ice country where temps at night normally go well below freezing and into the teens.
Therefore I get my winterizing done ahead of time in the low country with the RV anti freeze. For the trip to the cold country I carry a couple of 5 gallon containers of water plus some bottled water. That way you can still flush, etc.
Having learned the hard (cold) way that winterizing on the cold end of the trip is really harder and just downright miserable. What small amount of water that is flushed and drained from sinks does not cause a problem for me. You will have to judge your own situation on how much is in the tanks.