For a party that claims to love and respect the US Constitution, being filled to gills with "constitutional orginalists," I don't think any Republican has ever actually read the document. If they had even glanced over it, they would have looked at Article 6, Paragraph 2 also known as the "Federal Supremacy Clause." It's meaning is that law at the Federal level supercedes State laws and even state constitutions. The "Founding Fathers" intended for the Federal government to hold supremacy over the state governments... now it isn't written in plain English, but it is plain enough. Further, the end of our Civil War the court case Texas v White made it plain as day legally that no State may leave the union. Again, the party of "law and order" comes in clutch with that misunderstanding of what those words mean.

All that being said, succession isn't actually that popular (even in Texas). There would likely be a mass revolt against it in any state attempting it. Further, even a play at attempting it would be a political death sentence (as well as, perhaps, a literal one). Any insurrection against the federal government, even by a collection of contiguous states, would face a similar challenge the capitalists/monarchists (or Whites) of the Russian Civil War faced. The major population centers wouldn't recognize the authority of those in revolt and all they'd be left with would be the hinterlands and rural towns. Which is not a great position to be in strategically (again, see: the outcome of the Russian Civil War).

This is all not to mention, while States have their own National Guards, the insurrectionists would have to convince those troops to fire on troops who are wearing the same uniform. You'd have to convince their officers to break their oath to uphold both US law and face a possible death sentence if captured. I don't know Abbott's relationship with his National Guardsmen, but I highly doubt many officers will go in with him in this piece of political theater.