Okay, since things are getting very personal (again), I’m going to take a swing at answering the actual question as posted by the OP - assuming, as stated -
Thus I get to sidestep all the Civil War and divestment issues that are making people crazy. It happened, somehow, and some compromise that made everyone equally unhappy was worked out.
So, and as pointed out by @The_Librarian - the closest recent example of such a thing would probably be Brexit, and I think it’s a reasonably close analogy.
The Texan state would initially kind of lumber on, with inertia taking care of a lot of things, but all of a sudden would be subject to border restrictions, import/export laws, visa requirements, and the like. Larger, national corporations currently headquartered in Texas would have to consider if all the perks that Texas, notoriously business friendly, offers is now offset by all the new issues above, and I suspect a number would find it not worth the effort.
So, while oil refining and production will provide a source of income, I believe the economy would sustain an overall loss, which would increase dissatisfaction within the new state. Which would mean that the New Texas Republic (NTR from now on, -snicker-) would have to increase it’s support, probably by further doubling down on it’s base as it has before. So, the next step would be things like requiring all government and business do English only (long a dream of White Nationalists across the Southwest) or other, similar measures to cater to their own.
Which would further alienate any remaining moderates in the NTR, further restrict the economy, but would likely prove to their people that “things are getting done”. Meanwhile, Texas will go further down the Putin road - the corruption of the likes of Paxton, already legendary, will be fully protected, the remaining wealth will be concentrated in the hands of the few oligarchs, but they’ll be able to keep their base focused on how ‘Old America’, ‘those people’, and their own ‘fifth column’ are keeping the bright future they promised away, but soon (two weeks!) everything will start to turn around!
So the same things that are happening right now in Texas, but with the speed turned up to about 10x.