But in shorefront stretches of Staten Island and Queens that were all but demolished, and in broad sections of New Jersey and Long Island, gasoline was almost impossible to come by, electricity was still lacking and worried homeowners wondered when help would finally arrive.

Drivers in New Jersey faced 1970s-style gasoline rationing imposed by Gov. Chris Christie, while in New York, Gov. Andrew M. Cuomo said 8 million gallons had been unloaded from commercial tankers and another 28 million gallons would go into distribution terminals over the weekend. He also said the Defense Department was sending in 12 million gallons of fuel to be pumped from five mobile stations.

“They’ll have a 10-gallon limit,” the governor said. “The good news is, it’s going to be free.”

Only about 5,800 people in Manhattan awoke to find that they still lacked power, and crowds streamed into parks that reopened on a blindingly bright Saturday morning. Horse-drawn carriages were circling the roadways in Central Park again, and the grandstands were still in place for the New York City Marathon, though Mayor Michael R. Bloomberg on Friday had canceled it for the first time in its 42-year history.

But in many places that the storm pounded in its relentless push into the Northeast, there was a profound sense of isolation, with whole towns cut off from basic information, supplies and electricity. People in washed-out neighborhoods said they felt increasingly desperate. “I just keep waiting for someone with a megaphone and a car to just tell us what to do,” said Vikki Quinn, standing amid a pile of ruined belongings strewed in front of her flooded house in Long Beach on Long Island. “I’m lost.”

Hundreds of thousands of homes on Long Island were still without power Saturday, with temperatures expected to get down into the 30s overnight, and frustration with the utilities, particularly Long Island Power Authority, continued to rise.

“LIPA, get your act together,” Edward P. Mangano, the Nassau County executive, posted on his Facebook page Saturday. “This response and lack of communication with customers is shameful.”

In Long Beach, David O’Connor, 44, had begun to use his living room chairs as firewood. A neighbor, Gina Braddish, a 27-year-old newlywed, was planning to siphon gas from a boat that washed into her front yard. Older people on darkened streets have been shouting for help from second-floor windows, at eye-level with the buoys still trapped in trees.

“I am screaming mad because this is an inhumane way to live in the highest property-taxed area of the entire state,” said Hank Arkin, 60, a photographer in Merrick. “They had days of notice before the storm and nothing was done.”

President Obama held a briefing at the headquarters of the Federal Emergency Management Agency that included a conference call with Mr. Cuomo and Mr. Christie, along with Gov. Dannel P. Malloy of Connecticut. Mr. Cuomo said later that the president gave cabinet secretaries assignments to see that federal resources reached hard-hit states. And Mr. Cuomo, at a briefing in Lower Manhattan, said the city was moving forward.

“We are getting through it,” he said. “The worst is behind us.” But he also cautioned, “This entire situation is going to go on for a while.”

Mr. Cuomo said four subway lines that tie Manhattan to Brooklyn, the Bronx and Queens — the 4, 5, 6 and 7 lines — returned to life on Saturday morning, with four others — the D, F, J and M lines — set to begin running by nightfall. The Q train was also expected back by the end of the day on Saturday, and the 2 and 3 trains on Sunday.

But the L line remained flooded on Saturday — “wall to wall, ceiling to ceiling,” according to the chairman of the Metropolitan Transportation Authority, Joseph J. Lhota. A tunnel for the G line was also flooded; G trains were not expected to go back in service for several more days.

Utility crews from across the country struggled with a power network that had been battered. As they went from town to town and block to block, they trimmed trees and freed cables that had toppled in winds that approached 80 miles an hour. Despite nonstop work, the numbers were daunting. In New Jersey, Public Service Electric and Gas still had more than 600,000 customers without power on Saturday. Hoboken remained the biggest challenge because of extensive water damage, officials said.

Mr. Cuomo said that in New York, 60 percent of those who lost power in the storm had had it restored, but that 900,000 were still in the dark. On Long Island, where 1.2 million people lost power, about 550,000 had their power back by Saturday morning.

In Midtown Manhattan, riggers went to work high above West 57th Street, near Carnegie Hall, where the storm broke the boom on a construction crane and left it dangling 74 stories up. They hand-cranked the boom closer to the partially completed building, a condominium and hotel complex, and planned to strap the boom to the structure. Once that process was completed, the surrounding streets, which had been closed since the boom snapped in punishing winds on Monday afternoon, could reopen.

In the crane’s shadow in Central Park, the finish line for the canceled marathon was still in place, and the park and the side streets nearby were teeming with runners from all over the world. Some, like Gabriela Rose and Katharina Dawes of Hanover, Germany, said they had managed to continue their training during the week even though the park had been closed.

But outside the city, suburban residents still faced daunting challenges. Mr. Christie’s plan for odd-even rationing left some motorists confused. At an Exxon station on Route 440 in Bayonne, police officers and drivers standing in line argued about how to interpret the governor’s order, which read: “If your vehicle’s license plate ends in a letter (A, B, C…), you are only permitted to fuel the vehicle on odd-numbered days.”

The problem, the motorists said, was that all license plates in New Jersey end in letters except for vanity plates, so it seemed that nearly everyone could buy gas on Saturday but that no one could on Sunday.

“It’s an executive order from the governor’s office,” said the deputy police chief in Bayonne, Drew Niekrasz. “We have to follow it, even though it makes no sense.”

A spokesman for Mr. Christie acknowledged the confusion and agreed that only a small number of license plates ended with a number. The spokesman, Kevin Roberts, said the last number on a license plate would determine the day, even if letters came after it in the full alphanumeric sequence. He also said that vanity plates were to join the odd numbered group.

In Massapequa, on Long Island, drivers lined up at gas stations well before dawn, some leaving their cars as placeholders in the hours-long lines. By Saturday, some Suffolk County residents had given up and were traveling to Westport and Fairfield, Conn., to fill extra gas cans.

The authorities estimated that as many as 100,000 homes and businesses on Long Island had been destroyed or badly damaged in the storm, from bedroom communities in Nassau County to the towns of the South Shore to Long Island’s notable summer refuges — Fire Island, the Hamptons, Jones Beach — which were ravaged by the storm. Sand dunes were flattened and whole rows of beach houses crushed. The storm’s furious flood tide created new inlets that could become permanent parts of the topography.

“Fire Island is changed forever,” Steve Bellone, the Suffolk County executive, said at a news conference.

As Hurricane Sandy approached, Long Island appeared to be well out of its path. But the storm’s incredibly wide, counterclockwise swirl of damaging winds and rain, combined with an unusually high tide, sent a huge storm surge along its top like a right hook, slamming both the north and south shores of the island.

Many communities along the coast felt as if they were marooned, with cellphone reception spotty and power showing no signs of returning. Floods washed out roads, and the winds scattered cars across the beach, foiling the owners’ hopes of leaving.

Neighbors have been left to trade rumors about which hospitals were accepting patients, which streets had been hit by looters and whether the water was safe to drink.

An outsider who stopped to visit a darkened house in Long Beach was peppered with questions about the outside world. “The lines of communication have been very bad,” said Lincoln Jawahir, 49, who was among a group of neighborhood men who stayed behind to repair their homes by day and guard them by night.

And for those who made it out in search of supplies or news, confusion reigned. “People are on line but they don’t even know what they’re on line for,” said Lou Safonte, an information-technology engineer in Melville, where gas station lines stretched for blocks.

By the weekend, much of the shock and fear of the first few days had given way to anger.

Officials there said Friday night that power had been restored to 619,000 customers, but that it would take more than a week to restore service to hundreds of thousands of others. Senators Charles E. Schumer and Kirsten E. Gillibrand visited Long Island on Friday to survey the area and said the Army Corps of Engineers would be helping with the cleanup. The tour helped ease concerns that Long Island was being overshadowed.

“I’ve never seen such damage like this,” Mr. Schumer said after surveying Lindenhurst, one of the hardest-hit towns. “Never.”

Local officials have been struggling to help, but without better coordination with county and state authorities, Martin Oliner, the mayor of the village of Lawrence, said there was little he could do. “I can’t understand why in the last four days, until today, I have been having conversations that haven’t been meaningful,” Mayor Oliner said.

Reporting was contributed by Taylor Adams, Ruth Bashinsky, Matt Flegenheimer, Elizabeth A. Harris, Angela Macropoulos, Ray Rivera, Liz Robbins, Marc Santora, Nate Schweber, Stacey Stowe and Bernard Vaughan.