I live in San Antonio and hunt often on public land. Texas has both walk-in public hunts and drawn hunts. A lot depends on what you want to hunt and whether you’re willing to drive a few hours — as others have said, there are lots of opportunities for bird and small game hunting, but far fewer for big game hunting. Still, they’re around if you know where to look, and I’ve killed deer and hogs on public walk-in hunts. I’ve been much more successful with drawn hunts, but you have to apply for a lot of hunts and be patient while building up points. In recent years I’ve been drawn on average for about one hunt per year.
You’ll need an annual public hunting license in addition to your regular hunting license. The good news is that public hunting in Texas is much more simplified than other states.
Deer leases are expensive, but you can also get (relatively) inexpensive day hunts, and you can shoot hogs and exotics all year round.
DM me if you end up moving or if you have more specific questions.