In Mississippi, forecasters said 8 inches (20 centimeters) of rain had fallen in parts of Jasper and Jones counties by Monday, with several more inches possible.
Barry's centre is moving from northern Louisiana into Arkansas.
Though the storm weakened to a tropical depression on Sunday afternoon, its rain bands still created a flood threat stretching from central Louisiana to eastern MS and beyond.
Edwards says he's "extremely grateful" Barry did not produce the devastating floods that forecasters predicted.
John Bel Edwards encouraged inhabitants to be wary as they wandered outside following an end of the week in which many had sheltered indoors.
"So we continue to stress what we said yesterday, and that is not to let your guard down and become too complacent too soon; remain vigilant and pay attention and watch the advisories that come out", he said.
While the harshest part of the storm had passed Louisiana and moved on to Arkansas by Monday, parts of Louisiana were still suffering from pockets of heavy rain left in the aftermath.
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"This was a storm that clearly could have turned out differetly, in an unexpected way", he said.
More than 90 people were rescued in 11 parishes, but there were no reports of weather-related fatalities, Mr Edwards said. An additional 3 to 5 inches was expected Monday in the western part of the state, including parts of the rural Delta that have been flooded for months from the Mississippi River and its backwaters.
He's calling on the federal government to build pumps to drain water from the confluence of the Yazoo and Mississippi Rivers.
Businesses and state offices will reopen Monday in New Orleans, Mayor LaToya Cantrell said on Twitter. New Orleans was one of the hardest areas hit, already suffering from flooding in the days before Barry's full force hit the city. However, New Orleans levees appear to have held.
Across southern Louisiana, Barry's heavy winds scattered tree branches and knocked down power lines, leaving an estimated hundred thousand people without electricity.
The Amtrak Crescent was to leave NY for New Orleans on Monday, with service from New Orleans to NY resuming Tuesday. Westbound service resumes Saturday.
Barry's rains could have been much worse, said Edwards, adding that the state is now better prepared for future storms.