The first wave of data from states like Florida and North Carolina shows preliminary signs that Mrs. Clinton was building a slight edge even before the revelation that Donald J. Trump had bragged about sexual assault roiled the race.

Democrats are requesting more absentee ballots in Florida than they were at this point in 2012, with increases of 50 percent in the heavily Hispanic areas around Miami and Orlando. In North Carolina, where Mitt Romney built enough of a lead in early voting four years ago to edge out a victory over President Obama, Democrats are requesting mail-in ballots in larger numbers than in 2012, while Republicans’ participation is declining…

Barring a huge shake-up in other swing states, Mr. Trump will lose the race if he loses Florida. Florida Republicans have significantly narrowed a voter registration gap with Democrats, which stood at more than half a million people four years ago. As of mid-September, it had shrunk to just over 274,000. Republicans were also increasing their numbers in North Carolina, Nevada and Pennsylvania.

“We’ve really moved the margins in so many of these states,” said Chris Young, the national field director for the Republican National Committee. “And the reality is the Democrats are out of time when it comes to voter registration. In a majority of states, they’ve got a week, maybe two, tops.”