Texas is something like 98% private land. There are a few public hunting area but they have strict rules and are very crowded.

There is some BLM land in Eastern New Mexico but I'm not sure about the New Mexico hunting regulations. I'm just now investigating them for elk.

There are some antelope around Muleshoe and there are quite a few south of Clovis. There is some public land mixed in with large private ranches there but not sure on the access or how to get a tag. I know you can pay $$ and get a landowner tag very easily, and they aren't real expensive (less than $2,000 for the tag and hunting access). I've just never looked into it. You would have better luck in New Mexico than Texas for antelope in my opinion.