Gainesville, Fla. — It was homecoming weekend for the Florida Gators, and everyone seemed relieved to trade one American blood sport (politics) for a less psychically poisonous one (college football). But the election crept in, thanks to inspiration from Donald J. Trump’s vulgar boast from 2005. A hand-painted banner hung in front of one frat house read, “Grab Mizzou by the TIGER.”

Three weeks before the presidential election, both campaigns have been zipping around Florida, from Pensacola to Miami, vying for the state’s precious 29 Electoral College votes. Polling here remains tight in the race between Hillary Clinton and Mr. Trump, but Democrats seem to have built up a ground game advantage. In 2012, President Obama won Florida by about 74,000 votes. Since then, the Florida Democratic Party has submitted 177,000 new voter registration forms, compared with fewer than 13,000 forms submitted by the state Republican Party.

Last Friday, Stafford Jones, the chairman of the Alachua County Republicans, sat in a Gainesville volunteer office while his small dog snuffled around his desk. He said that pollsters’ turnout models might not factor in previously disengaged voters who will be voting for Mr. Trump.

“It is not out of the realm of possibility that everybody that thinks that Hillary Clinton has got this sewn up is completely wrong,” he said.