THE PLAINS, Va.—For clues about the strength of both parties’ messages heading into 2024, look no further than this suburban Northern Virginia enclave, where Democrats and Republicans are betting big on a hotly contested state Senate race.

Typically, such local elections don’t have a major impact outside their direct communities. But this one could decide control of the Virginia legislature and Republican Gov. Glenn Youngkin’s ability to enact conservative policies on abortion, taxes and education—making it closely watched in the run up to next year’s presidential race.