Take US 287 from Amarillo to Wichita Falls. Easy road, plenty of services along the way. Take US -281 south from Wichita Falls to I-20 then over to FW. The south loop is fine to get to I-35. (Going from Amarillo to Lubbock adds too many unnecessary miles.).
If you want to avoid FW completely, take TX 171 from Weatherford to Cleburne. Then US 67 to Alvarado.
Do not take US 287 into the northern part of Fort Worth. There is massive construction along I-35 from near where US 287 joins I-35 to near downtown. It blocks easy access to Loop 820 from I-35, making either the west or east loop a bad trip.
US-67 is a great road for an RV from Dallas to Alpine. Yes, it goes through some small towns, but wide lanes, good shoulders and many sections are four lane. Be aware that the speed limit on most of US-67 is 75 mph, as it is on many two lane roads in Texas.
Note - the West Bound iron bridge over the Brazos River between Cleburne and Glen Rose was hit by a dump truck this week and is closed. Traffic is two way on the other bridge so there are some delays. Estimates for repair / reopening are mid-June. This would impact you if you plan to stop at Tres Rios Resort for you first couple days testing of your new rig.