The polls made the race a close one, narrow advantage to Obama, as two weeks of back-to-back conventions approached. Hundreds of millions of dollars have been spent on television ads, with hundreds of million more to come, almost all of it airing in a small group of battleground states expected to settle the election.
The list included Florida as well as North Carolina, where the Democratic National Convention will be held in one week's time.
Scott declared a state of emergency earlier in the day as the storm approached the Florida Keys, more than 400 miles from Tampa. Forecasters said it was on a track to head west of the convention city, but predicted strong winds and rain at a minimum on Monday as the delegates were to board buses for their first trip to the hall.
"We are a hospitality state. We know how to take care of people and we want to ensure their safety," Scott said.
Apart from weather concerns, a heavy security presence was already in evidence. Miles of fencing were designed to create a secure zone around a tract of land that included the convention hall, the hotel where Romney will stay and a nearby convention center where journalists and others worked.
Obama did his best to intrude on the Republican unity tableau.
In an interview with The Associated Press, he accused Romney of holding "extreme positions" on economic and social issues, while pledging a willingness on his own part to agree to "a whole range of compromise" with Republicans if he is re-elected.
He did not elaborate, but his pledge seemed designed to appeal to independents and other voters who say they are tired of seemingly perpetual campaign bickering and Washington gridlock.
Plans for Vice President Joe Biden to campaign in Florida were cancelled, also because of the threat posed by the storm.
But Romney said Obama's entire campaign rested on his ability to persuade people to ignore his record and listen instead to his rhetoric.
"It is not his words people have to listen to. It's his action and his record," he said in his appearance in Powell, Ohio. "And if they look at that, they'll take him out of the office and put people into the office who'll actually get America going again."
Romney's speech included an appeal to women made on economic grounds rather than on the basis of social issues like abortion, the sort of approach the Republican hopes will eat into Obama's polling advantage among female voters.
"I want to make sure that we help entrepreneurs and innovators. I want to speak to the women of America who have dreams, who begin businesses in their homes, who begin businesses out in the marketplace, who are working at various enterprises and companies," he said.
Romney envisioned an economic resurgence fueled by abundant energy, expanded trade and a skilled workforce. If that happens, "America is going to surprise the world. We're going to stand out as a shining city on a hill in part because of our extraordinary economy," he said to the cheers of an estimated 5,000 supporters.
Romney's determination to turn the attention to the economy follows two weeks of controversy over Medicare, courtesy of Obama's campaign, as well as abortion, the result of a comment by Rep. Todd Akin, the party's candidate in a Senate race in Missouri.
Romney joined an unsuccessful effort by party leaders to force Akin to quit his race after he said women who are raped rarely become pregnant, a view unsupported by medical evidence.
He also fought back hard in recent days in person and television advertising against Obama's allegations that he and running mate Ryan would remake Medicare in a way that would undermine the health of future seniors.
AP White House Correspondent Ben Feller and Associated Press writers Steve Peoples and Philip Elliott in Ohio, and Brian Bakst and Suzette Laboy in Florida contributed to this report