Not necessarily. Might want to check out the National Park Service (yes you read that right) land around Lake Meredith. There is public hunting there and I would be surprised if they didn't have feral pigs.
As for exotics, the only public land hunt I know of for those that is not a draw is (once again) National Park Service land around Amistad Reservoir near Del Rio, TX. The only realistic exotics they have are Mouflon sheep. I've killed one there and it's a good hunt when they are on the NPS property, but they are easily run off onto adjacent private. They are supposed to have Auodad too, but I've never seen one.
Other than those two areas, everything else I know of in West TX or the panhandle is by draw only. Of course, the National Forests in East Texas have plenty of feral pigs, and they aren't that hard to find if you know what you're doing.