Are we talking about marriage licenses? If so, residency doesn't matter. Where the marriage is performed applies. If the wedding was performed in Florida, then the license is obtained there and after the ceremony, filed with the clerk which becomes the official marriage record. You get one certified copy free, but I'd recommend getting several because you may need to show (or in some cases give) a copy to various people to show your change in status and your name change if you chose to do that.
You need not file or register anything in Texas. Of course, as a Texas resident, you will have a Texas Driver's license. If you changed your name as part of the marriage, you bring the Florida marriage certificate to the DMV and they'll update your license (and car registrations if necessary). Similarly, you will likely want to inform the SSA and if you have a passport, you have to send in a DS-5504, your marriage certificate,and your old passport to get a new one.
There are no other real requirements, your obviously should tell your employer, but you can just put your new name on the tax forms without having to explain anything to the IRS or the Texas Comptroller.