I make the trip from Yuma AZ to Tampa FL at least 4 times per year. Sometimes in 40 ft RV sometimes in a car. I drive Baton Rouge to Ft. Stockton in a car in one day and if in RV I stop in Kerrville TX. Ft. Stockton would be my target if I were you coming from Houston.
Do fill up in Ft. Stockton as there is nothing until El Paso.
Interstates are safe - no worries. Cell Tower connecting to Mexico is a non-problem with Verizon anywhere along route.
Las Cruces NM has a really nice RV resort right off I-10 if you can make it that far.
Big thing to watch out for is fuel. Don't let it get to low as it can be some distance between stops.
Enjoy the trip.