Here is a route for you to consider. Certainly the one I would take:
AT Texarkana take US 59 (sign would say "Houston")
At Linden, TX take TX 155 all the way to Tyler
Take Loop 323 around West side of Tyler
Stay on TX 155 to Palestine (loop around West side of Palestine)
Take US 79 in Palestine
You could go all the way down to Round Rock north of Houston on US 79
or you could cut over to I-35 on US 84 just west of Palestine. I always stay on US 79, but you do have a choice. Otherwise you pick up I-35 at Round Rock.
I've made this route many times.
Now, I'm going to express a purely personal observation: for some reason I've always found Austin to be one of the worst cities to tow through. I'd far rather go through Houston or around Dallas on I-20 than through Austin. So I'd probably skirt on over to Burnet on TX 29 and then take US 281 on into San Antonio, but this is just my personal choice and I would enjoy the drive much more.
If you have any question about any part of this route, I'll be happy to respond.
Let me add: the TX 19 would also be fine. You'd have no trouble with that one. Good loop around Athens and I just came over TX 31 from Corcicana to Athens (actually Tyler) last week.