Do you want to stick to Interstates the whole way? Once you get west of I35 in Texas, there are many fairly lightly traveled quality US and even state highways.
Houston is pretty nasty, yes. You could take I49 to Shreveport, then go through Dallas and Ft. Worth on I20. I20 runs along the southern edge and so avoids most of the DFW metroplex traffic. After Ft. Worth, you could stay on I20 and meet back up with I10, or you could take US 84 through Lubbock and up to I40. Clovis NM is the slowest spot on US 84. US 287 from Ft. Worth to Amarillo is also a good route.
If you want to avoid all the big cities, take US 84 to US 380 (before Lubbock) and take that to I25 in NM, then jog north on I25 to US 60, taking that into AZ. Once in AZ, take US 180 and hook up with I40. Slowest spot on that route is Roswell NM, but Roswell isn't that big and won't delay you much. I've been that way several times, and it's a good route.
Another fun road I've taken a few times is NM state highway 9 right up against the Mexican border. They finally improved the signage somewhat. Used to be very difficult to find the El Paso end of that road. Now it's only a little obscure instead of being nearly impossible. There is just about _no_ one on that road. And it's pretty good as far as being straight (the worst spot has a few 50 mph curves) and level. It follows an abandoned railroad line. At the western end, take state highway 80 (formerly US 80) west through Douglas, AZ and on until it rejoins I10. I've also tried I8, to AZ 85, to avoid Phoenix. Or stay on I8 into CA and take CA 79 to I15-- seemed about the least painful way to slip into the LA area. I've tried CA 62 north of I10 too, and it's fine until you get within 20 miles of I10, then it's terrible. CA 18 and CA 138 over to CA 14 are crummy too-- too close to LA to escape the traffic.
I recommend against US 82 in NM. The western end of that road is not the best quality, being narrow, twisty, steep, and bumpy.
I also prefer I40 to I10. On rare occasions, the border patrol will stop I10 to check for illegals. Finally, California seems to think it's a sovereign nation and at the border will want to hassle you over taking any agricultural products, plants, or animals into the state.
If you go up I-49, make sure to stop in Natchitoches at Lazyone's for meat pies. Ooo, mamma. By the way, being in Louisiana, Natchitoches is pronounced Nack-uh-dish, and Lazyone's rhymes with Jazzy bones.