One thing to take into consideration besides the snakes, bugs etc is the heat. Most of the Southern/eastern US is being hit by a heat wave. 100+ degree weather + pavement/asphalt etc can burn the hell out of your feet. A friend of mine in high school had to have skin grafts because she walked barefoot across a parking lot. If you are doing mostly trail stuff the heat may be less of a factor but then you have the snakes etc to take into account.
That said, I do not see there being much chance of a spider biting you while walking/running. They seem to bite people while they are sleeping (if it is the legs/body) or while moving things (bites to the hands and arms). A snake might get you if you surprise it but given how rattlesnakes etc strike (above the ankle) anything short of boots might not make a difference. So...barefoot or tennis shoes...seems about the same to me.
Hopefully others who actually live in TX will chime in.