Every state is different in how they do this.
In my state (Georgia) we do not register to vote by any party affiliation.
During the primary season, any registered voter can vote however they wish.
Upon arriving at your polling place,
you ask for either a republican or a democrat ballot.
In the event of a run-off election, you cannot switch parties.
(you can’t participate in choosing the quarterback for both teams)
Seems there is likely a transgender joke in that statement..... but we'll let it go.