This comes from the Lubbock Avalanche-Journal (an ironic name for a newspaper in Lubbock, if I ever heard one. Lubbock=flat as pancake).
The civil division chief of the Lubbock County District Attorney's office says that Lubbock County will begin to issue marriage licenses on Monday, July 6. They will be opening at 8 am. Offices are closed today - July 3, for the holiday.
This is fairly significant in that Lubbock County, with about 300,000 residents, was the most populated county in the state that was not issuing licenses.
The city of Lubbock is in Lubbock County. The main campus of Texas Tech University is in Lubbock. As an aside, TTU has perhaps the most awesome college sports live mascot ever. It's a person dressed in a red Zorro outfit riding on a jet black stallion.
Still going through the newspapers and press releases from around the state trying to figure out status of each county. Might have to wait until Monday for further news, though.