Make sure you change your address. You can put a fraud alert on your credit agency files. You can also purchase one of the identify theft packages.
Have you filed for divorce? If not, I would ASAP. In Texas, there is no "legal separation."
Obviously, you are concerned but no one has a crystal ball and can predict what will happen.
I would go to your local social security office to discuss the matter directly with them. You should also monitor your credit through the 3 major credit reporting companies. This can be a free report.
If you really want to just avoid any issues and are that concerned about it, I would suggest talking with the local field office about changing your social security number.