Laura,
We make that trip almost annually. Staying on I10 is best. Only traffic concerns will be Houston, San Antonio, and El Paso, none of which are bad if you hit them after/before work traffic. You can bypass San Antonio by exiting at Seguin TX and taking TX 46 until you reach I10 in Bourne TX.
We always stay in rest stops or Wal-Marts. Mississippi has an excellent rest stop near the Stennis Space center. It's not too far from the AL border (exit 60+_). It has a separate RV area past the truck parking where you can pull off the road into an area with separate picnic tables for each slot. Louisiana only has one rest stop and it's about midway between MS and TX (last time we were through). TX rest stops are universally OK to Good. West TX (beyond San Antonio) has them spaced out about every 75-90 miles or so. They do tend to fill up at night (we usually drive till 9 or so) so if you see a parking slot when pulling in, take it. If you can park along the curb and avoid the diagonal parking you may be able to use one of your bedroom slides. Wal-Marts are also good. I usually go in and ask permission so they know I'm there.
Have fun! Steve