Another place to consider if your heart is set on Texas is Wichita Falls in far N. Texas. I lived there off and on for 10 years and at first I thought I was stuck in the isolated flat-lands of hell, but it grew on me. The people were very nice. You could raise kids there without fear. There were plenty of stores and restaurants but it still had a 'smaller town' feel to it. It has an airport (basically flies you to Dallas/Ft Worth for connecting flights, but still...) There were great places to shoot (shotgun, rifle, handgun). And the leases were somewhat affordable. A trip along Hwy 287 toward the Panhandle got you to more hunting land opportunities (hogs, deer, antelope, mule deer.) A trip south along Hwy 281 took you to the Hill Country, with more hunting opportunities. Lake Texoma wasn't too far away if you are into fishing and recreational lake activity. If you ride motorcycles, AR was a six hour ride to "good" roads.
As previously stated, Texas property taxes will hurt you. Hunting lease prices will hurt you. And the only "backpack" hunting I can think of would be in W. Texas (rugged country) and you STILL have to pay for land access. Nearly all land is privately owned. You pay to play, which is getting more true everywhere. As stated, Austin is a big "no!" for a hunter like you. It is the progressive part of California plopped right into the middle of Texas. Homeless people, traffic, and ridiculous virtue signaling laws abound there. Just say "no" and look elsewhere.
If you like trees, look at East Texas (The Piney Woods.) It borders Louisiana and there IS public land to hunt in Louisiana. Hmmm...maybe that's where you should be looking. Just some more thoughts to consider. (Did I mention you might want to look at OK, MO, or AR instead?)