More than 20,000 people along Florida's Gulf Coast have ordered to evacuate, and many have – boarding up their windows before getting out of town.
And even once that's done, plenty of folks are still worried.
"Can't get flood insurance, not worried about hurricane, worried about the water," one resident of Cedar Key, Florida, told CBS News correspondent Cheryl Casone as Alberto drew near.
Officials still worry that some residents of low-lying areas prone to flooding may not be taking the storm seriously.
"A lot of people aren't going to leave," said Jackie Gorman, Cedar Key's community development director. "We're hoping this is going to be a small one, but who knows."
At 5 a.m. EDT, Alberto was centered about 65 miles west of Cedar Key, and was moving northeast at about 9 mph toward an expected landfall around midday near where the Florida peninsula juts out into the gulf. The National Hurricane Center says the top sustained winds are at 65 mph; the minimum for a hurricane is 74 mph.
A hurricane warning was posted for the Gulf Coast and a tropical storm warning was issued from north of Daytona Beach to the Georgia-South Carolina line.
"We want to make sure people take this, very, very seriously," Florida Gov. Jeb Bush said Monday, as he signed a declaration of emergency allowing him to call up the National Guard and put laws against price gouging in place.
"We don't want to overdo it. It's not a Katrina or a Wilma, but storm surge and flooding could still cause loss of life," said Max Mayfield, director of the National Hurricane Center.
The biggest impact of the day may be from a storm surge – predicted to be between six and ten feet high – expected to hit between Cedar Key and Apalachee Bay around 8 a.m. or 9 a.m. when Alberto's center moves over the coastline.
Forecasters say Alberto may become a weak Category 1 hurricane, but probably no stronger, because the warm water a hurricane needs for fuel isn't too deep in the area where it now looms.
CBS News Meteorologist George Cullen says the heaviest rain – probably 2 to 4 inches - is likely in Georgia and the coastlines of the Carolinas, with most of Florida getting only an additional inch or so and some areas staying rain free.
Parts of the Sunshine State have already got quite a soaking: about 3 to 5 inches in Tampa and Sarasota, and 2 to 3 inches across the northern half of the state. That's great news for northern Florida, notes Cullen, since that area has been dealing with drought conditions.
"We were surprised, but we've been surprised before," hurricane specialist Richard Pasch said. "The center in disorganized storms can re-form and jump."
Forecasters said it could bring 4 to 10 inches of rain to central Florida and southeastern Georgia. Rain already was falling Monday and at least two tornadoes had formed, though there were no immediate reports of any injuries or damage.
In Florida, homeowners gassed up their vehicles and stocked up on chain saws, plywood and other emergency supplies. Workers at a marina in St. Petersburg said they planned to work through the night securing more than 600 boats.
"This is a little earlier than I expected," said marina manager Walter Miller. "But we've had a bad couple of years, so it's not entirely unexpected."
Alberto also prevented the crew of space shuttle Discovery from flying Monday to the Kennedy Space Center from Houston for several days of dress rehearsals for their expected launch in July.
Wal-Mart Stores Inc. ramped up its emergency operations center Monday for the kind of disaster relief effort that won it praise for responding faster than the government last year after Hurricane Katrina.
On Monday, Alberto drenched western Cuba after a weekend of heavy rains prompted evacuations, flooded tobacco farms, caused some dilapidated buildings to collapse and flooded low-lying areas in Havana.
Scientists say the 2006 season could produce as many as 16 named storms, six of them major hurricanes. Last year's hurricane season was the most destructive on record and the busiest in 154 years of storm tracking, with a record 28 named storms and a record 15 hurricanes.