Satellite images showed large areas under water, including the Grand Bahama International airport and the town of Marsh Harbour on Abaco Island.
Dorian fell in strength on Tuesday to category two, but the northwest islands continue to endure heavy rain, high winds and storm surges.
Five people were killed when the storm hit the Abaco Islands.
Residents of the northern Bahamas, some trapped on roofs, sent out pleas for help as the storm thrashed the islands on Monday night, stalling at category five with 185mph winds.
The eastern US coast remains on alert for the hurricane.
Queen Elizabeth, the head of state of the Bahamas, said she and Prince Phillip were "shocked and saddened to learn of the devastation caused by Hurricane Dorian" and she sent condolences to families and friends of the victims.
"At this very difficult time, my thoughts and prayers are with those who have seen their homes and property destroyed and I also send my gratitude to the emergency services and volunteers who are supporting the rescue and recovery effort," she said in a statement to the country's governor-general, Sir Cornelius Smith.
What's the latest on Dorian?
The most recent update from the US National Hurricane Center (NHC) placed Dorian just north of Grand Bahama, home to about 50,000 people, having moved barely 20km (12 miles) in a day.
The BBC Weather service said that having remained stationary for many hours, the storm was now moving at 1mph (1.6kph) with maximum sustained winds of 115mph.
But the NHC warned the storm was still producing higher gusts at 140 mph, with storm surges of 10ft-15ft (3m to 4.5m) above normal, and the agency advised residents to remain in shelter on Grand Bahama throughout Tuesday.
The winds at the core of the storm were spinning so fast that the centre of the storm was collapsing on itself, causing it to expand and damage a larger area, according to the BBC Weather service.
Steve McAndrew, of the International Red Cross, told the BBC he had been involved in rescue operations for 20 years and could not recall a hurricane ever being listed as stationary.
Palm Beach county in Florida - less than 100 miles to the west - saw gusts of up to 60mph on Tuesday.
Meanwhile, the UK has sent a team of three humanitarian experts to help assess damage caused by the hurricane, the government said.
A Royal Fleet Auxiliary ship, equipped with a helicopter and loaded with relief supplies, is also on standby if needed by the Bahamian authorities, a statement added.