If the registration is mailed it can be post-dated on the 4th and still be acceptable. So if you drop it at a post office about 6pm on Feb 4th is the deadline.
You just have to be registered to vote or is there some separate registration for the Texas GOP? I already have my voter registration card and on the Texas GOP website it says that "In Texas, registered voters may vote in either political party’s primary; however, a voter may vote in only one party’s primary in each election cycle." so I thought I was ok, but maybe not?