AFAIK - not American - there’s one primary election for each full election. It’s not like you can vote in the Republican primary, then go down the street to the Democrat polling booth and vote there - they are the same polling station. So pick one.
As others point out, many states make you register as one or the other, so buying two party memberships does not gain you anything. (Don’t need to be a paid-up member in the USA, just declare your leanings as I understand).
The other point is, as I understand, you are voting the full gamut. The USA is legendary in Canada as the place where even dog-catchers are elected. By selecting a political party choice, you are allowed to primary vote for all positions at once for that one party (maybe even in a butterfly ballot?)
So selecting one party to vote for the worst presidential candidate, means you are not able to vote for a good candidate from your favourite party for congress or senate, where all the work is (not) done. Or mayor, governor, state legislature, bridge lane-closing committee, school board, etc. Or dog-catcher.