Hillary Clinton has established a slim edge over Donald J. Trump in early-voter turnout in several vital swing states, pressing her longstanding advantages in state-level organization and potentially mitigating the fallout from her campaign’s latest scrap with the F.B.I.

Even as Democrats continued to reel from revived questions about Mrs. Clinton’s use of a private email server as secretary of state — a jolt delivered 11 days before the election in an abstruse letter from the F.B.I. director, James B. Comey — turnout tallies and interviews with dozens of early voters suggest that even a vintage “October surprise” may pack less of a punch than it once did.

In a race between two deeply polarizing candidates, opinions appear to have been cemented weeks if not months ago for most voters. And the contest is well underway in some of the most important battlegrounds.

At least 21 million people have voted so far across the country. In the states that are most likely to decide the election — among them Florida, Colorado and Nevada — close to a quarter of the electorate has already cast ballots. While their votes will not be counted until Election Day, registered Democrats are outperforming Republicans in key demographics and urban areas there and in North Carolina, where extensive in-person voting began late last week and which has emerged as one of the most closely contested battlegrounds for the White House and control of the Senate.