I'd look at the ST as three parts: desert southwest, Texas, and the Gulf Coast. Either way, you're going to hit Texas in the middle, so that doesn't help you decide.
Going west, you'll hit hot and humid into mid-October along the Gulf. Good news: you'll miss most of the 90 degree/90 % humidity. Bad news: it'll be 80/90 days instead. There's not much relief from the humidity at the end of the day. I'd expect the desert to be getting chilly by the time you get there, but it's easier (at least for me) to add layers to deal with cool than it is to generate enough sweat to stay alive in hot, humid weather.
Going east, you'll have some toasty days in the desert near the start, but it's a dry heat, and you say you're used to that. You'll have longer days so you can make some miles in the evening if you need a shady siesta in the early afternoon. By late October into November, the weather will be pleasant in the Gulf, it's likely to be a bit cooler and drier. Unless a hurricane is coming.
Either way, be careful about riding into sunrise and sunset (that means try to avoid it!). The glare makes it difficult for motorists to see you.