One small problem, there is no independent party.
In Texas, there are several main ways for a voter to affiliate with a party: by being accepted to vote in a party’s primary election, by taking the required oath at a party precinct convention, or by taking a party oath of affiliation generally
You cannot change your party registration in Texas in a given year once you have declared a party. You can only wait until the year is over and then select another party if you wish. If you don't select a party, you have NOT changed your voter registration because you have no voter registration on Jan 1.
A voter’s affiliation with a party automatically expires at the end of each calendar year, which is December 31. (§162.010).
If you haven't affiliated with a party in a calendar year, that doesn't change your voter registration to independent. As far as the state is concerned you are still able to declare a party affiliation any time in the calendar year. If you could actually change your voter registration to independent then you would be ineligible to vote in either party's primaries but the state says you are still entitled to do so since you haven't declared a party yet.