The Interstates will mostly be open all winter. Storms may stop traffic for a day or two in which case travellers can be stuck somewhere waiting for the road to be cleared. Getting off the Interstates is the difficulty. I have no idea how your rig handles on snowy and icy roads.
I-20 south is snowbird country there are a lot of parks in NM/AZ and CA for that matter in that area. Death Valley prime time is winter/spring for example. Chihuahua NM is a great place to explore but their CG is limited to small RV's. There are a lot of state parks too.
You need to be flexible, watch the weather to the north it could be a warm spring after all. Weather is, as always, unpredictable. But Big Bend from what I've heard has their peak number of visitors around Thanksgiving-Christmas. But snowbirds winter down there too. Terlingua is pretty much closed up in the summer but is fully in business in winter. There are three of four RV parks in operation then too, not counting the ones in the NP.
I'd write off the Texas shore this winter after their bad weather experience.