Are you all looking at Texas only? The deadline for applying in Wyoming is May 31. You'll need PP to get a good buck tag though. Some are drawn in random every year.
Doe tags are cheaper and random draw. You can apply for 2 doe tags in initial drawing then 2 more and a leftover buck tag in the leftover draw in June.
Fishing and grouse hunting are excellent in fall, spawning trout and sage grouse and dusky grouse hunting ,after the doe tags are filled. Maybe get a leftover deer or elk tag to go with the antelope tags.