Dem SC primary was not conducted on the same day.
True, but once you vote in one of the primaries, you are legally ineligible to vote in the other.
link:
VOTER REGISTRATION
Registration Deadline: You must be registered at least 30 days prior to any election in order to vote in that election. Registration by mail applications must be postmarked at least 30 days prior to that particular election to be eligible.
Party Affiliation: South Carolina does not have registration by party. Voters may vote in either party's primary, but not both.