I also say you should plan on dropping down through Texarkana and on down to I-10, then across. You might luck out and be able to take the northerly route but I would not count on it.
It's colder in NM than in the AZ valley. Las Cruces often gets below freezing at night. Unless you really want to take on the challenges of below freezing camping in a TT, I'd figure on the AZ valley.
If you take the above advice on the route, I'd expect to be able to travel the whole way un-winterized, with water on tap. But you never know.
