Republicans have dominated redistricting. Here’s why that could change.

State-level politics matter a lot to voters’ lives, from new laws to the drawing of political districts. While “down-ballot” races tend to get little attention, this year some significant rebalancing of power in state legislatures could occur.

Georgia residents Larry and Helene Hunt – an African American couple who vote straight-ticket Democrat – have figured out over the years that “change comes slow here in the South,” as Ms. Hunt says.

Their 14-year tenure in Richmond Hill, a suburb of Savannah, exactly matches the years that Republicans have had a...