Texas
I was in your situation 2 years ago except it wasn't military but it was still for work. Lived in Colorado and got sent to the DFW area. I had heard rumors about the lack of public land but I never figured it would be as bad as stated. Boy was I wrong. Something like 98% of land in Texas is private. Their are public hunting areas but they are far a few between and get absolutely pounded by hunters. As stated in an above post hunting in Texas is a big money business. Everything is leased or charge a high ( for me anyway) price for a day use fee. I have lived in the west my entire life and had more public land to hunt than I could imagine. This whole concept has been a huge culture shock to me. I'm surprised 7-11 doesnt charge you to lease the gas pump when you fuel up. It is normal down here cause that is how it has been done for many years, younger generation doesnt know any other way. There are places you can take your own birds or buy birds from them and train your dog but again it is costly.
With all that being said my advice is this. Look for a quail lease. These are typically with other guys who you would be on the lease with. The amount of people depends on the size of the land usually. These leases vary greatly in quality and price. 90% of the quail leases will be after the deer season which leaves you almost two months left in the quail season. Deer hunters will pay quadruple per acre than a quail hunter so that is the reason they are leased to deer hunters then quail hunters. There are quail only leases that go all season long but they generally run big dollars $15k - $20k per hunter. All the leases are priced per acre, average is about $2-$3 an acre so a 2000 acre place with three hunters is going to be around $2000 per hunter for the almost two months of the season. You need to dig deep to find what works for you and something that is close. Most leases offer a place to camp during the lease. Some of the leases offer year round access that allow you to hunt varmits and pigs throughout the year and certain deer depending on the management plan of the group. Usually there is a person in charge of the lease that you deal with and that person deals directly with the land owner. All concerns, complaints, etc go to the person in charge and it gets handled there. It is important to make sure you get a long with the group of people you have a lease with.
South Texas has a lot of leases which sounds like where you would be. West Texas is a hot bed for quail also. It stays pretty warm year round. I trained some dogs today and had to cut it short due to the heat, south Texas is usually warmer. Also another thing that was a change for me was worrying about snakes during the hunting season. Snakes hibernation period around here is very short and sometimes not at all. Can't get used to seeing snakes out at the end of december. Seems the last couple of years are banner years for copper heads and rattlers and I hate snakes. Another piece of advice would be not to skip any heart worm or tick meds. Only bringing this up as it was a learning curve coming to a warmer year young climate.
I dont want this to be a Texas bashing thread. It isn't all bad. The people are nice, the weather is great but the hunting is all pay to play and that takes some getting used to.
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