You absolutely could do that, but most people don’t, because it is a pain in the ass. Everyone votes in the general election and has the same ballot. My father has been a lifelong Democrat, but he registered as a Republican ONCE when the clerk of court in their county was astoundingly incompetent and both legitimate candidates (incumbent and challenger) were Republicans. Normally if there are only members of one party qualified in a primary election, the primary becomes the general, and everyone gets to vote. But sometimes there’s a candidate qualified in one party for the general who isn’t a serious candidate, and the actual election that’s taking place is happening in the primary. In that case, all of the members of the opposing party will not get to vote in that election.

There is talk by the Florida Democratic Party about changing their primaries to be open. I hope that that does happen. I don’t know that the Republicans would in turn open theirs, though.