For what it's worth, in Texas riding along an interstate is legal unless otherwise posted. I cannot recall ever seeing a sign prohibiting it, but if there are any stretches where you cannot cycle, they would be in the big cities. There should be no prohibitions in West Texas. Certainly, if the AdventureCycling map shows routes on the interstate, you can be confidant that it is legal to ride there.
You can always ride on the interstate frontage roads, if there are any. Out in West Texas, I believe you will find many long stretches that have no frontage roads.
That being said, certainly riding along any interstate is not the most pleasant riding. An alternate route that avoids interstates would probably be more attractive. Keep in mind though, that towns can be many miles apart out there, so plan your route carefully and always make sure you have plenty of water. There are also lots of place names in Texas that show up on maps even though there is little, if anything, remaining. Make sure that any vital resupply town has someplace to resupply!