Tell me exactly how the people who want to secede are going to go about it. They all get together and say we're seceding or words to that effect? So the rest of the state including the authorities with guns are just going to say alright, you win? How are they going to do it? Be specific.
OK, I'll be specific. This isn't based on any kind of special knowledge of the laws involved, just my take on a logical way for it to happen.
1. Those Texas citizens in favor of secession from the Union make their case, and get a referendum put on the state ballot so all the citizens of Texas can vote on it;
2. If the referendum to secede passes, then the State Legislature would be instructed to petition the Federal Government to consider our request to leave the Union;
3. The Federal legislature (House and Senate) would then consider the request. Now here's where it gets murky, because they could just refuse to consider it. What I would prefer to see would be that they move the request forward in a way similar to how an Amendment to the Constitution would proceed, i.e., a bill allowing Texas to secede would be voted on in the House and the Senate and if passed would then go to each State's legislature asking them to vote on Texas' petition to secede, yea or nay. If whatever majority is established (simple majority, 2/3 majority, etc.) agrees to it, then Texas leaves peacefully. If not, then Texas stays in the Union.
What would happen if Texas asks to leave and is not allowed to leave is a different question.
What happens to Social Security and other Federal entitlement programs if Texas secedes is simple: they are a product of the U.S. Federal Government and not the Republic of Texas Government, and so Texas as a nation would have no responsibility to continue providing them. If people want to continue receiving such, I imagine they would have to maintain their U.S. citizenship; in my perfect scenario, anyone wishing to become a citizen of the Republic of Texas would be required to renounce their U.S. citizenship. I haven't researched it, but I'm guessing that anyone renouncing their U.S. citizenship loses their Federal entitlement program benefits. Anyone who maintained their U.S. citizenship but continued living in Texas would be considered foreign residents, and would need visas etc. In effect, they would be the same as any other U.S. citizen who chooses to live outside of the borders of the U.S.
Again, that's just what seems logical to me. Your mileage may vary.