Hey Agu, welcome and good details here.
For Houston, I’d start with the community colleges first. They tend to be the best value, and a lot of them will prep you for an AWS test without blowing past your $2–3k budget. If you want something faster, there are private welding schools too, but the quality is all over the place so I’d vet them pretty hard.
A couple things to sanity-check before you pick a program:
• Do you want an actual AWS cert test (D17.1 structural, pipe, etc.), or do you just want solid TIG training and a school certificate?
• What kind of work are you aiming for in Houston: structural, pipe, refinery, shop/fab?
If you tell me what side of Houston you’ll be on (or a zip) and what type of welding you want to end up doing, I can point you to 2–3 programs that are actually worth calling first, plus what to ask them so you don’t get sold.