“If we’ve got all these people running around bumping into one another, then we’ve got a lot of trouble,” Mr. Villere said. “The challenge is management.”

In both states, analysts say runoffs might favor Republicans because Tea Party backers and Libertarian voters are more likely to line up behind a Republican than a Democrat. But those voters might also stay home. Fewer people vote in runoffs than on Election Day, and experts say the outcome would rest heavily on which party did a better job of getting its backers to the polls.

Both races would test how much the states have changed: whether Louisiana has moved so far to the right that it votes out the last sitting Democrat elected statewide, and whether Georgia’s shifting demographics have moved the state far enough to the left to elect a Democrat to the Senate for the first time in nearly two decades. In Georgia, history is not on the Democrats’ side. Republicans have won all five of the runoffs there since 1992.

Image David Perdue, a Republican, is vying for a seat in Georgia. Credit... Kevin Liles for The New York Times

“Democrats have a harder time of getting people to show up regularly in the midterm election, and this is a subset of that — a midterm runoff election,” said Charles Bullock, a political scientist at the University of Georgia. “What Georgia Democrats have tried to do is put together something like the get-out-the-vote effort that the Obama campaign has used. So it will be a real test for them if they have to get those voters back for a runoff.”

Georgia’s unusually long nine-week Senate runoff would last through the holiday season. Strategists in both parties are already expressing concern about how to keep exhausted voters engaged — a concern that appears well founded.

“I’m pretty much sick of everything,” said Alecia Britt, 37, an administrative assistant, after she voted Friday in Morrow, Ga., south of Atlanta. She said she had little appetite for another round of campaigning and nasty advertisements. “It just leaves a bad taste in your mouth when you hear the mudslinging.”