In St. Mary Parish along the central Louisiana coast, officials were in "watch and wait" mode, according to Duval Arthur, the parish's emergency preparedness director.
"The way this storm is moving, so slow, it's really kind of hard to make any plans, to start any plans ... it's changed on us actually three times," he said.
Officials are planning to issue street-flooding warnings in recreational areas located on the water, he said. "As this storm gets closer, we're going to have 3-to-4-foot higher than normal tides."
Louisiana's Terrebonne Parish and Lafourche Parish also issued precautionary states of emergency Sunday.
BP said Sunday it was closely monitoring the progress of Debby and that all the company's oil and natural gas production in the Gulf was to be shut down by midday.
"Helicopters will continue evacuation of offshore personnel as weather conditions permit," BP spokesman Brett Clanton said. "Those unable to evacuate will shelter in place for the storm."
Nine oil and gas production platforms have been evacuated of the 596 manned platforms in the Gulf of Mexico, the federal government said Saturday. One of 70 rigs was evacuated.
A tropical storm warning means that tropical storm conditions, including sustained winds of 39 to 73 mph, are expected in the warning area within 36 hours, said CNN meteorologist Bonnie Schneider.
"If you are under this warning, take your precautions now," she said. "Tie down or bring inside any furniture, toys or decorations you keep outside. They may blow about in strong winds and cause damage or injuries.
"If you are in a low-lying area at risk for flooding, know the best route to higher ground where you live," Schneider said, and she urged people to stay abreast of local weather warnings.
A tropical storm watch means tropical storm conditions, including winds of at least 39 mph, are possible within the watch area within 12 to 24 hours, the hurricane center said.