There are PLENTY of places to hunt in Texas. TP&W has all the info you need for public hunting:
http://www.tpwd.state.tx.us/huntwild/hunt/public/They have maps of the public land for year-round hunting, as well as info on the "lottery hunts" they put on throughout the year. I've done these hunts several times, and you really can't beat it.
As far as "private land". Texas is well know for thier deer leases. They can be anything from small time $500-$1000 dollars per gun per year with a local farmer, to $5000 and up on the "name brand" ranches in the hill country and down south.
If you want to hunt a private lease...which I prefer.... you can get a good lease if you're willing to do a little work. Most leases in Texas are advertised within a month of deer season. By the time the ad is out in the paper, the good leases are already taken. Here's a secret.... go to the Austin American Statesman (the big Austin newspaper) and ask to search classified from September through October of the last season. You'll be able to find some of the leases that were advertised before the new season ever starts. THAT WAY, you have a chance to do some scouting and possibly get in some dove or quail hunting.
Other options like talking to folks at the feedstores, small time gun shops or even small-time cafes in the more rural suburbs might pay off. Another option is to get to know folks at the cattle auction barns....this can be a little awkward, but it will pay off if you try.
Finding a lease in Austin is going to be tough if you wait until the last minute. Most folks in that area have a LOT of money, and have caused the prices to go insanely high.
Another option is Ft. Hood. They allow unguided public hinting during bird season and archery season. It's less than an hour up the road, but they have taken some really good deer off of the training area in the past 10 years. Here's a link:
http://www.hoodmwr.com/sportsmen'scenter.htmGive Hunt Control or the Pro Shop a call and they will help you out.
Other hints.... stay away from San Antonio for dove hunting. There are a TON of idiots throwing shot everywhere.
If you're looking for good hunting, try the hill country. Anywhere within the Lampassas-Junction-Interstate 35 triangle is good hunting with the chance for hogs and free-range exotics. You can call the Chamber of Commerse in most of the towns and ask about hunting leases. They can be pretty helpful.
If you have any other questions... there are enough of us that hunt/hunted Texas that can help you out.