Until they secede and wipe out the second biggest sector of their economy which is defense/military which bring in outside dollars to the state. Once the defense contracts are cancelled and the military bases closed so outside dollars no longer flow in, even if Texas tries to keep some facilities open it will be funded by internal Texas dollars which means a huge economic hit in terms of "balance of trade".
The of course you have the impact on the shipping industry for Texas ports that current receive cargoes for distribution to other States in the Union. If Texas becomes it's own country a large part of those shipments would be to other gulf ports (like Mississippi, Louisiana, Alabama, and Florida) for follow on shipment to the states. Goods shipped through Texas means that all those goods become foreign imports from Texas with additional duties to be paid.
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Definately would be interesting to see how "punished" Texas would be becomming a nation that imports and exports, but I doubt there would be much difference between the existing 3 way NAFTA and one that would include The Republic of Texas.
Not sure where the reference to the military being the "second biggest sector of their economy" comes from, however, let's assume its true. I'd expect the USA would need Texas as an Ally as much as Israel, or Egypt, or Germany, and would continue their military presence PLUS contribute billion$$ in military aid.
As for the decline in social security and medicare, I see no problem with The Texas Republic allowing its citizens to immigrate into the USA for free visits to the emergency rooms and to enjoy the same social services illegal immigrants receive now, taking jobs, and sending money to relatives in Texas. US citizens would of course enjoy their visits to Texas where pharmaceuticals would be sold at Canadian rates.