I got into whitewater paddling when I was kid in Austin, TX... so take it from me, it can be done!
It would be a drive for you, and I don't know what it's running right now, but the Guadelupe and San Marcos rivers are kind of the best bet around central Texas. Don't know if you know about it, but if you're looking for instruction there is a great little kayak school in San Marcos. Here is the link: index.htm
As far as a step up, in my opinion, your best bet would be to take a long weekend and head up to the Rio Grande. Base yourself out of Taos, or there is good camping along the river, and run the Racecourse section. At the typical lower flows it's easy access roadside Class III. Parts of it get up to Class IV with the spring melts and then there is hopefully also the Taos Box which is a gorgeous III / IV wilderness day canyon. The Rio Grande is only about a 12 hour drive from your area.
Please feel free to contact me if you head out to New Mexico. I lived and paddled there for years and can give you any kind of beta you want.