Gov. Ricardo A. Rosselló of Puerto Rico said on Monday that the island was on the brink of a “humanitarian crisis” nearly a week after Hurricane Maria knocked out its power and most of its water, and left residents waiting in excruciating lines for fuel. He called on Congress to prevent a deepening disaster.

Stressing that Puerto Rico, a United States commonwealth, deserved the same treatment as hurricane-ravaged states, the governor urged Republican leaders and the federal government to move swiftly to send more money, supplies and relief workers. It was a plea echoed by Puerto Rico’s allies in Congress, who are pushing for quick movement on a new relief bill and a loosening of financial debt obligations for the island, which is still reeling from a corrosive economic crisis.

“Puerto Rico, which is part of the United States, can turn into a humanitarian crisis,” Governor Rosselló said. “To avoid that, recognize that we Puerto Ricans are American citizens; when we speak of a catastrophe, everyone must be treated equally.”

And Mr. Rosselló did not mince words about the potential impact on the mainland, where Puerto Ricans are expected to arrive in droves to escape the post-Maria hardships they will face on the island, including a shortage of already hard-to-find jobs.

“If we want to prevent, for example, a mass exodus, we have to take action. Congress, take note: Take action, permit Puerto Rico to have the necessary resources,” Mr. Rosselló said.

Residents face obstacles in navigating almost every step toward a normal life, with little hope of dramatic progress anytime soon.

At Petroamerica Pagán de Colón, an independent living apartment building in San Juan for people over 62, residents have made do with limited water and 14 floors of stairs to climb for crucial goods. Those who are disabled or too sick to climb depend on neighbors to get them food and water. Some have not been able to bathe.

“People were abandoned for seven days,” said Alejandro Melendez, a resident. “There were sick people on the floor, thrown there.”

In the coastal city of Arecibo, where water remains in short supply, residents gather around spouts to collect rain as they peer anxiously down streets for water deliveries. “They are not giving us anything, not even hope,” said Cannabis Angel Nebot, 43. “At least, come around and give us hope, even if it’s a lie.”

Mr. Nebot and his girlfriend, Ixia Milly Rivera, spent the weekend driving around their neighborhood trying to find water to clean off the residue of seven feet of mud Hurricane Maria left in its wake. They could not find any at City Hall. They did not fare any better at the emergency operations center, which is reserving its water for people in shelters or with special needs.

“I have one water truck; I need 10,” said David Latorre, Arecibo’s emergency management director. “It was an odyssey to find food. We had to break down doors to get it. The food system collapsed.”

But Mr. Latorre was still optimistic. “I know FEMA will come,” he said.

Republicans in Washington pushed back forcefully on Monday at any suggestion that the relief effort for Puerto Rico was less aggressive than it had been for Florida and Texas.

After facing criticism for a lack of public support for Puerto Rico, President Trump on Monday posted a series of tweets that tied the natural disaster to the island’s already fragile economic situation. He said that while Florida and Texas were coping well with hurricane damage, “Puerto Rico, which was already suffering from broken infrastructure & massive debt, is in deep trouble.”

“Its old electrical grid, which was in terrible shape, was devastated,” he continued. “Much of the Island was destroyed, with billions of dollars owed to Wall Street and the banks which, sadly, must be dealt with. Food, water and medical are top priorities — and doing well.”

The White House rejected criticism of its response.

“The federal response has been anything but slow,” said Sarah Huckabee Sanders, the White House press secretary. “In fact, there has been an unprecedented push through of billions of dollars in federal assistance.”

In a visit to the island on Monday, Senator Marco Rubio of Florida made assurances that Congress understood the gravity of the situation. “Our commitment is to make sure that Puerto Rico will recover stronger than ever,” he said.

Congressional leaders said on Monday that they are now awaiting assessments of the damage in Puerto Rico, as well as a formal disaster request from the Trump administration, before they can act. A request is not expected until early- to mid-October, according to senior congressional aides.

But Representative Rodney Frelinghuysen, the Republican chairman of the House Appropriations Committee, said on Monday that Puerto Ricans on the island “are entitled to equal treatment under the law.”

The Federal Emergency Management Agency is drawing from the same $15.3 billion pot of money that was approved this month by lawmakers in response to Hurricane Harvey, which hit Texas, and Hurricane Irma, which hit Florida and damaged Puerto Rico and the United States Virgin Islands.

The director of the agency, Brock Long, and Thomas P. Bossert, President Trump’s Homeland Security adviser, were both in Puerto Rico on Monday to assess the damage and Mr. Rosselló’s priorities.

Mr. Long said that the federal government had 10,000 people “working around the clock” to help Puerto Rico. Money is not the issue at the moment, he said. More pressing, federal officials say, are the obstacles to smoothly delivering water, food and other essentials to a devastated island.

San Juan airport’s airfield is damaged. And while six ports have opened, most are operating on a limited basis. Many roads remain cluttered with debris, and bridges must be checked. Getting around and finding housing for federal workers remains a challenge. And supplies and workers can’t just roll in on convoys from the next state over. The nearest state to Puerto Rico is more than 1,000 miles away by sea.

Still, Mr. Long said, ships and planes are streaming in with crucial goods.

“We’ve got a lot of work to do,” Mr. Long said at the news conference Monday. “We realize that Maria was 1 mile-per-hour from being a Category 5 storm, but it’s the worst Puerto Rico has seen. It’s been very complex for us to respond, from a logistical nature of the island.”

Reinforcements were arriving every day, officials said, including fuel, military personnel and law enforcement agents. The Puerto Rico Ports Authority said shipping companies were also sending 1,000 shipping containers with water, medicine, generators and other supplies.

Puerto Ricans can now file damage claims with FEMA, which has sent teams to 10 municipalities to go house to house to collect information and pass it on, the governor said. More roads are also being cleared to bring food.

With 60 percent of the island without water, and all of it without electricity, Puerto Rico’s frustration is palpable across the island.

Elí Díaz Atienza, the executive president of Puerto Rico’s water authority, said the agency is running water with generators it had in stock. All of the island’s waste water and water treatment plants lack electricity. “We still haven’t received the ones that FEMA is going to give me, but they are working with us,” he said. “We need 2,500 generators for the entire system to be running on generator power. Obviously we are not going to find that.”

Progress is being made, slowly. And congressional Democrats have teamed up with some Republicans to push hard for faster relief.

“The situation is desperate,” said Senator Chuck Schumer, the Democratic leader, on the Senate floor. “Puerto Rico has taken a serious punch to the gut. They need our help. They need it now.”

Some Democrats want Congress to quickly approve a relief bill but also to temporarily forgive Puerto Rico’s loan repayments and remove a requirement that Puerto Rico make a contribution into the federal emergency pot, a cost-sharing arrangement that is typical for disaster-affected states.

But Puerto Rico has little clout in the Capitol. As a commonwealth, it does not have a voting member of Congress, only a resident commissioner, something that has long shackled the island financially and politically.

“This is what makes Puerto Rico a beggar,” said Edwin Meléndez, the director of the Center for Puerto Rican Studies at Hunter College. “Our citizenship is second class. We don’t have the same rights as other Americans. If we move to the states, we have those rights. In Puerto Rico, we don’t.”

With the top floor sheared off Mr. Nebot’s house and a swollen river a block away, he and his girlfriend say they may move to the mainland. Everything they own is doused in brown sludge.

“Do you know what people are referring to this town as? You know ‘The Walking Dead’?” Ms. Rivera said, noting that crime was sure to rise as people grew increasingly desperate. “We are afraid for our lives.”