Once you get to Stony Brook, talk to the exchange office there about doing a year at a uni in Texas as part of your Stony Brook degree. Stony Brook is part of the national student exchange program. You’d normally pay your NY State Stony Brook tuition, and be able to attend one of the listed schools in TX for up to a year:
http://www.nse.org/exchange/memcam.asp
In addition, you’re welcome to do things like internships over the summers in Texas. This gives you even more productive time in TX, as you pay the lower, in-state rate on tuition in NY, but start to set up a life for yourself in TX. Then, when you graduate Stony Brook, you move to TX.
If you do decide to transfer to a school in TX, then even if you did take a gap year, you still wouldn’t qualify for the in-state rate on tuition at public unis in TX. The thing is, if you’re under age 25, it’s about where your parents live as much as it is about where you live, and your father doesn’t live in TX. So even if you move there, take a gap year, get a job, etc., you still won’t get in-state tuition.
That said, if you do decide to transfer to a school in TX, apply both to public and privates. See where you get in. See what aid each offers you. Calculate what you’ll need in loans for each. Then decide where you’ll go, or if perhaps you’ll even stay in NY. While it is the case that publics in TX will make you pay the out of state rate, that doesn’t mean they still won’t be cheaper than the privates in TX. You won’t really know until you apply and etc. So apply to both types of schools, should the time come.