Michael Magnanti listened in his church in Oxford, N.C., on Sunday morning as the pastor asked whether anyone had an announcement for the congregation.

Mr. Magnanti stood from the choir loft, his voice warmed up from hymns.

“Election Day is happening on Tuesday,” said Mr. Magnanti, the chair of the Granville County Republican Party. “I’m begging you to get out and vote, because this is the most important election if you’re alive today.”

It was a final, nearly desperate plea to voters in a battleground state where the presidential race was still too close to call. Polls released over the weekend, including surveys by The New York Times and Siena College, suggested that the race between Donald J. Trump and Vice President Kamala Harris was agonizingly close in swing states from Pennsylvania to Michigan to Arizona, offering little relief to Americans who are looking for resolution near the end of a long, exhausting campaign.

The uncertainty has also provided fuel. In the last hours of the campaign, candidates, staff members and tens of thousands of volunteers across the country were in an all-out sprint, racing to pin down as many voters as possible. The specter of an evenly divided race made the push from both sides to get out every last voter even more urgent and frenetic.

On arena stages and in storefront campaign offices, party leaders exhorted crowds to keep going. Volunteers set off in pairs with clipboards and kept knocking on doors, even in the rain.

They were all determined to use the time they had left to boost the turnout for their candidate over the top, perhaps making a tiny difference in an election where that could be enough.