Ummm, you are not going to be riding through ANY desert, given the route you have marked on your site. Certainly there's not a *thousand miles* of desert ANYWHERE in Texas.
Your route (as marked) goes nowhere near Amarillo (mentioned on your site), which is the panhandle (the part of the state that sticks up under the narrow part of Oklahoma). The Panhandle, of course, is *prairie*, not desert. The desert, and there is some, is in the southwestern part of the state, starting more or less in Big Bend National Park. You aren't going there.
The best you can hope for is a little cactus around Austin (but not desert, cactus is a weed in the middle south part of the state).
Contrary to popular belief, there is darn little desert in Texas, and none of it is in Dallas, Austin, or parts in between. You're basically on the wrong side of the state (which is to say, the wetter side). Fort Worth is only about 50-60 miles from Dallas, but it's considerably drier. The further east and north you go, the wetter it is. Houston, of course, gets rain almost every day, and is very green, being in the eastern part of the state, and practically on the coast.
The route you have marked until Austin is HIGHLY populated, humid, and parts of it somewhat hilly, especially Austin, which is a bear. You're going to be dealing with *traffic*, not lack of facilities. You need to be worrying about *HUMIDITY,* not the desert. Dress accordingly. And remember that the only part of Texas that likes cyclists is Austin.
Given the route you have marked, you're going to be going down (approximately) IH35, which is essentially the "great divide" of Texas, dividing the wetter east from the drier (but not yet desert) west. The route is loud and noisy, but rideable, mostly because we have service roads on either side of every major freeway. Picking up 281 in Waxahachie (south of Dallas) will be much prettier and rideable, and there are small towns spaced appropriately. Whether there is camping, I don't know. Though frequently without shoulders, 281 is usually pretty low traffic north of Austin, very shady, beautiful route. (closer to San Antonio, it turns into a dangerous, high-speed, divided roadway, but it also has wide shoulders there). If you want motels, and I would in early September, for sure, you should probably stick to IH35, since the smaller towns may not have hotels, and certainly not newer hotels. Or, go online and look for B&B's in the area, some parts of 281, like Marble Falls, attract tourists.
Will it be hot? Yep. Maybe. Depends on when you're riding through here. If you're here in late September, it will still be hot and humid during the day, about 80F and humid at night, especially as you get further south (Austin is *much* warmer than Dallas, because Dallas is on the prairie, but Austin is not). But starting the middle of October it'll all start cooling down. Heck, two years ago, Austin got *snow* in December; Dallas regularly gets ice storms around xmas.
Will parts of the trip be empty of people? Yes. I would say most of those parts are in the eastern part of the state as you go towards Houston. There's really not much between Bastrop and Columbus but some very small towns. But you will start seeing more trees after a while. But between the time you enter Texas, and the time you leave Austin/Bastrop, you're going to be hard put not to run people over with your bike. (or rather, get run over). Then, between Columbus and Katy, there's basically nothing, including trees. They start to disappear, as you are on the edge of the coastal plain. There aren't even many services for cars, but there are some little towns off IH10 that might have little grocery marts or something, I've never stopped between San Antonio and Houston except in Columbus.
Hope some of that helps.