The final clause of Article VI is not in perpetuity. That oath only applies if you are still part of the union. If you separate from the union, then you would no longer be held to that oath.
In marriage, you promise to be loyal to your spouse, and even take an oath “until death do us part”. However, if you get a divorce, you would no longer be held to an oath of loyalty, nor would anyone expect you to. To separate, the parties involved negotiate a settlement, which is what would ideally occur after secession. Forcing a state to remain in the union would be like forcing an abused spouse to stay married. The federal government is in clear violation of its “marriage contract”.
However, if you wanted to get technical, Texas secession from the United States was done by delegation, which is how it will likely be done again. People (on a county-by-county basis) will vote to select delegates for a secession convention. These delegates will decide whether to secede. This takes the state legislature out of the picture, if you were so concerned with Article VI.
Circular reasoning.
In practice, if an oath-bound group of men decides to separate from the union, they're guilty of insurrection, suppression of which is explicitly authorized by the Constitution. If a state officer "shall be bound" by his oath to support the Constitution, and as such he agrees to the superiority of the Federal to the state government. That's the whole point of the government defined by the Constitution! And by pulling out of the union... he's breaking his oath.