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Let's start by saying one of the great myths of winter travel is that heading south is a good way to avoid snow or ice. Every single cross country route sees winter weather at some point of the year, and places farther south are often less able to deal with it when it hits. Unless there is a storm in the forecast for the specific days you'll be traveling, going with a direct route is usually the best bet. That gets you to your destination fastest, meaning the fewest possible days to see a storm.

The route I'd be looking at first would be I-64 to West Virginia, US-35 to Dayton, I-70 to Indy, I-74/I-280 to Davenport, I-80/I-680 across Iowa, I-29 to Sioux Falls, and then I-90 the rest of the way to Washington. I-90 is actually one of the lowest elevation routes across the Rockies, and at times it can be too cold for heavy snow in February, so it actually has some advantages in winter travel you might not otherwise think of.

This is mostly a direct route, but with the exception of a short stretch through WV, it is toll free, and with the exception of a short stretch of US-35 in Northern WV, it is 4+ lanes all the way. The trip is just under 3000 miles, and at a reasonable and steady pace of 500 miles a day while towing, it will take you 6 days. That still gives you 3 extra days in case you need to slow down or even stop for a storm and wait for conditions to improve.