There are tons and tons of year-round and season leases available in Texas. Most hunting is done that way here. There is some public land in East and West Texas, but I've never pursued it much so others would have to chime in. I'd say that most people hunt private leases, because there is so much more private land than public.
I've got a brother in law in Spring, so I know just where you're talking about. While there's lots of deer country here, the two main hunting areas are South Texas (the area around Webb county (Laredo) and the surrounding counties) and the Hill Country (Llano, Mason, San Saba county area in west-central Texas). A $1,000 should get you a nice year round or season lease in the Hill Country. I hunted 12 years on a 2,000acre lease in Llano w/10 other hunters for $850 a year, then moved to a lease in San Saba with 10 hunters for $1,000 a yr. and hunted there about 5 years. I don't have a lease now, but my old lease is now $1,100 a year. They're out there if you can find them. I've shot lots of good deer, hogs, turkey, etc. on those leases. The Hill Country has the highest concentration of deer in the state, maybe the country. Not as many big bucks as other areas, but it all depends on what your definition of "big" is. I've shot many 8 pointers with 16 - 18" spreads. The biggest bodied deer I've seen on any of our leases probably went 150#, with 90-125 lbs being more typical of the bucks. Does go maybe 80 - 100 or so usually in that area. Most counties in that area are 5 deer limits, 2 bucks and 3 does, but it varies.
South Texas is more of a high dollar affair, but much much bigger bucks. I couldn't afford a season lease there probably, but if you can, go for it, that's where the big daddies are usually taken. There are LOTS of hogs in south Texas, you may want to go on a weekend hunt or something down there for them, its usually only a few hundred dollars and it's an adventure.
Of course, that's all generalization. There are lots of places to hunt all over, you just have to find them or hook up with somebody that has a lease and can tell you when they have an opening. I envy the states with the public land at times, because it'd be cheaper to hunt. But then again, I don't have to wear blaze orange, I don't have to worry about some yahoo sitting in my stand, and you're probably not going to get shot at.
"As for the soutern comment, Texas is its own place that is different from the culture of say alabama, miss, Ga, Tenn. La. thats just my opinion."
You're definitely right about that!
Good luck, and welcome to Texas!