My winterizing strategy is to be south of I-10, usually in Texas.
Spent much of one winter at Davis Mountains State Park just outside Fort Davis - 40 miles south of where I-10 and I-20 join/ split.
The hill country west of San Antonio gets a few days below freezing each year. And a winter weather event - with a little ice - about one year in four. Farther west maybe one year in two has a bit of ice.
If you can watch local TV weather - they will shout and moan about the horrible winter weather coming, and being from MA, you are going to say - this isn't winter weather.
Normally, the ice has cleared and the temps are above freezing by 9-10 am. Watch the 18 wheelers/ big trucks. If they are parked, you should park also.
We did see a multi-day ice event - the first they had seen in seven years. TXDOT closed I-10 between Balmorhea and Van Horn for two days.
You will know if the roads are clear to leave San Antonio and head west. Check at Fort Stockton at truck stops - we use Flying J - they will know about the roads to El Paso. There are also several RV parks in Fort Stockton if needed.
We vastly prefer to travel US-90 from San Antonio to Van Horn rather than I-10 - it is only 20 miles longer distance wise, less than an hour longer time wise - and much less traffic. It's a bit farther south - but if there was a major winter storm predicted, I'd stay on I-10. US-90 goes through some long stretches with no cell service, no traffic, no towns, no homes.
While closer to the border, I'd have no concerns about traveling that route, though I wouldn't 'Wally Dock' as the roadside rest areas/ picnic areas. I'd prefer to be in an RV park for the night.
Several at Del Rio, good state park near the Pecos River, a fair overnight stop place at the Judge Roy Bean Museum in Langtry, good RV parks at Sanderson, Marathon, Alpine (Lost Alaskan on the north side if town is very nice), and Marfa.