ALBANY, N.Y. — With allegations of unsettling behavior toward women spilling into the public eye, Gov. Andrew M. Cuomo spent Tuesday fending off calls for his resignation, with few voluble defenders in a moment of unparalleled weakness in his decade-long tenure in Albany.

Signs of the governor’s diminished sway were everywhere.

A small, but expanding, coterie of Democratic lawmakers called on Mr. Cuomo to step down, as did the state Working Families Party, which has frequently clashed with the governor. Among some donors, there was an increasing sense of discomfort with reports of Mr. Cuomo’s behavior and uncertainty around his future, with one active Democratic donor describing a growing instinct to “hedge their bets.”

Representative Lee Zeldin, a Long Island Republican, announced on Tuesday that he was “actively exploring” a run for governor.

Still, for all that, one major bulwark to any forced departure — the Democrat-dominated statehouse, which could impeach him — appeared to be holding, for now at least. Impeachment would require mass defections by Democrats in both the State Assembly and the Senate, which seemed unlikely as of Tuesday.

The leaders of both chambers, however, did strike a deal to impose limits and additional oversight on Mr. Cuomo’s pandemic-era powers.

The move was the latest rebuke of the governor by members of his own party, a way for the Legislature to flex its muscle and reprimand him for withholding data on nursing home deaths and, more symbolically, the sexual harassment allegations.

Mr. Cuomo, 63, has been accused of sexually harassing two former aides, including Charlotte Bennett, 25, who said the governor had complained of being lonely and asked if she slept with older men during a meeting in his Capitol office last year.

Mr. Cuomo has not directly challenged Ms. Bennett’s claims, though he flatly denied some other accusations against him. On Sunday night, he sought to explain his behavior, saying that some of his comments in the past “may have been insensitive or too personal,” and “made others feel in ways I never intended.”

“I acknowledge some of the things I have said have been misinterpreted as an unwanted flirtation,” Mr. Cuomo said. “To the extent anyone felt that way, I am truly sorry about that.”

On Monday, The New York Times detailed another encounter where Mr. Cuomo had touched a young woman’s bare back, cupped her face and planted an unwanted kiss on her cheek at a 2019 wedding.

That disclosure prompted Representative Kathleen Rice, a Long Island Democrat, to call for Mr. Cuomo’s resignation, saying, “The time has come.”

Ms. Rice, a former Nassau County district attorney, became the first Democratic member of the state’s congressional delegation to call for the governor to step down. Others, including Senators Chuck Schumer and Kirsten Gillibrand, reiterated their support for an independent investigation.

Mr. Cuomo’s outsize presence in state politics seems to be cutting both ways in his current political crisis: Many privately express spite toward him, but few in his party have dared to take him on. The promised investigation of Mr. Cuomo’s behavior has given many elected officials political cover, allowing them to express concern about the allegations, while asserting that they are still, at this point, just that: allegations, albeit from three women who have given on-the-record accounts.

Mr. Cuomo has not made a public appearance since last Wednesday morning, when the first accuser — Lindsey Boylan — posted a lengthy essay about his behavior, including what she said was another unwanted kiss, on the lips, in 2018. The governor has denied her account.

Mr. Cuomo, a third-term Democrat, had no comment on Tuesday on the calls to resign, though he has previously voiced support for an investigation of his own behavior, granting the state attorney general, Letitia James, permission to deputize an outside lawyer to look into the claims.

When asked what the governor was doing amid intense scrutiny into his conduct, an aide referred to a news release about a pilot program to administer the new Johnson & Johnson vaccine at mass vaccination sites.

A number of state legislators, especially some of the governor’s most frequent critics, were far more direct than many of their counterparts in Washington as they pressed for Mr. Cuomo to step aside.

“I don’t think someone who treats women like that should be in power,” said State Senator John C. Liu. “Silence is complicity. And when women, and people, suffer harassment of any kind, they should not fear speaking out. Part of that means not allowing people to feel lonely and isolated by, quote, ‘playing it safe,’ end quote, and keeping silent.”

The deal to restrict Mr. Cuomo’s pandemic-era powers will still allow the governor to issue executive orders that are deemed critical to responding to the pandemic. Existing directives — such as those mandating mask wearing — would remain in effect. Lawmakers, however, would now have the ability to review any directives the governor decides to extend or modify, such as increasing indoor capacity in restaurants.

The Legislature, where Democrats possess majorities large enough to override a potential veto from Mr. Cuomo, could vote on the legislation as soon as Friday or Monday, according to Assemblyman John T. McDonald, a Democrat.

There are also signs that among some key political constituencies, there remains significant good will toward Mr. Cuomo, especially tied to how he led the state in the early months of the pandemic.

“I think it’s ridiculous to ask him to resign,” said Hazel N. Dukes, the president of the N.A.A.C.P. New York State Conference. “We have an investigation going on. Everybody should have due process for allegations made against them. That’s democracy in America.”

Ms. Dukes’s remarks were echoed by Jay Jacobs, a friend of Mr. Cuomo’s who serves as New York State Democratic Party chairman. He issued a lengthy statement on Tuesday saying it was “both premature and unfair for anyone to opine on the outcome until that investigation is completed and the results reported.”

But that sort of response was openly mocked by some of the governor’s critics, including a raft of younger, more progressive lawmakers.

“You know what’s premature and unfair?” wrote State Senator Jessica Ramos, a Queens Democrat, on Twitter. “Grabbing someone’s face without their consent.”

Indeed, the controversy over Mr. Cuomo’s behavior seemed to be revealing a split in the state party between established Democratic Party loyalists and more ideological, socially conscious members.

Laurie A. Cumbo, the majority leader for the New York City Council, said that it was vital to both “provide a safe space and place for women to come forward” and to allow an independent investigation to run its course.

“Sure we can cancel him, sure we can ask him to resign, sure we can demand that he resign and we move forward, but once we’ve canceled Gov. Cuomo, are we just creating, you know, this cycle?” she said. “How can we do something other than cancel here to really get to the heart of creating a solution and the understanding and the humanity that it takes in a workplace environment to address this issue?”

Many lawmakers were eager to reach a deal on the governor’s emergency powers, concerned that the debate had overshadowed negotiations of the state budget, due April 1.

But the fallout from Mr. Cuomo’s sexual harassment allegations has once again threatened to eclipse, or possibly derail, the budget discussions, an important moment of the year in Albany when the most significant policies are negotiated.

Indeed, the business community is already fretting about what a governor with reduced leverage might mean for their interests.

“For the last decade, they’ve relied on Cuomo as the moderating voice in state public policy and budget negotiations,” said Kathryn S. Wylde, the president of the Partnership for New York City, a nonprofit business group. “The concern is that his voice will be weakened and distracted during a critical period for the city and state as we’re attempting to get out of a deep economic hole.”