CW, might be good to review what the TexSOS says on VoteTexas.gov.
Their site seems to say the only restriction is you can not vote in both runoffs.
Texas doesn’t register voters by party. If you did not vote in the primary and are a registered voter you can vote in the runoff of either party. If you voted in the primary you can only vote in the runoff for the party in which you voted in during the primary.
Texas Election Code:
snip
Sec. 172.125. ADDITIONAL PROCEDURE FOR ACCEPTING VOTERS IN RUNOFF.
(a) For a runoff primary election, the voter registrar shall enter on the list of registered voters a notation beside each voter’s name indicating the preceding party primary for which the voter was accepted for voting, if any.
(b) An election officer at a runoff primary election polling place shall determine whether the name of a voter offering to vote is noted on the list as having been accepted for voting in another party’s primary. If the voter’s name is so noted, the voter may not be accepted for voting at the runoff unless the voter executes an affidavit stating that the voter did not vote in the primary or participate in a convention of another party during the same voting year.
Added by Acts 1993, 73rd Leg., ch. 728, Sec. 67, eff. Sept. 1, 1993.
http://www.statutes.legis.state.tx.us/Docs/EL/htm/EL.172.htm