As my name suggests I am somewhere in Texas...
I would definitely suggest it. The hunting is Fantastic, there are places all over Texas where you can hunt exotics (on game ranches of course) but there are wild populations of Blackbuck, Axis deer, Nilgai antelope, Audad goats, Japanese snow monkeys (believe it or not, there are some in south Texas), and there are rumors of a small population of warthhogs in south Texas.
Also we have your normal Texas big game such as white tail deer, mule deer, pronghorns (some), mountain lions, coyotes (I know, not really big game) and more hogs than you can shake a stick at. There is also plenty of wing shooting. We have Turkey, ducks, doves, pheasants, and quail. There is also fantastic fishing in both fresh water and saltwater.
However you asked about the weather. We have a saying here in Texas; if you don't like the weather... wait 5 minutes. It is 100% true. We do have hot weather here in Texas. No matter what part of Texas you live in you will have hot weather from April- September. But you can do something about the humidity. The closer to the east/north east the more humid it gets. but the farther you get from said directions, the drier it gets. You can live in south west/ west / northwest Texas where it is basically is a desert and you wont have to worry about the humidity. However, the farther west you go the less game you are going to find. (Because for the most part, Texas turns into dry scrub brush and rocks once you go about 100 miles past the hill country).
If you are worried are worried about having too much humidity but still want good hunting I would suggest finding a place down in south Texas. South Texas hands down has the best hunting for almost everything. And it aint nearly as humid as the Houston or San antonio area.