I have traveled 10/20 over the years during the time frame you indicated and yes the roads can be closed due to snow, and when I did it was in dallas eastward one year. I dont like 10 as its too far south and makes it a longer tiring trip particularly when going north to get to van horn. 20 is pretty much a straight shot and there are enough pull offs and rv parks to deal with most contingencies.
When doing any travel as you know, the weather determines what you do and we never ever really know what 'ma nature' is going to do for certain now do we? be vigilant is the main thing and enjoy as best ya can.