Having worked the elections for several years now, I can answer that question with some authority. You do not "register" with either party. In the Primary you will have to choose one or the other. You can also choose NOT to have your card stamped with the party you selected. You will be given a separate card, and your name will be marked as voted. You cannot vote in the other party's Primary or run-off. In the General Election in Nov. both parties will be on one ballot.
In Texas, you have to be registered with the Republican party to vote in their primary. You get to choose your GOP affiliation during the Republican primary or Republican primary runoff. Once you have voted in the primary election, you cannot change your party affiliation for the upcoming election.
htt-p://votetexas-.-gov Your State Secretary's website. It says nothing about Party affiliation affecting Primary voting (either way) but the url has a contact the Secretary of State's office link. They will have the best and latest information for you. Although some people are going to ask why you would care unless your planning a party raid... (I disabled the link with horizontal slashes.)
you don't have to be registered but you can only vote in one of the primaries either republican or Democrat. after you vote your card will be stamped with rep or dec so you dont go try to vote in the other primary.