ALBANY — The airwaves are virtually silent. The fiery criticism of years past has given way to conciliatory press releases. And the halls of the Capitol ring not with angry protests but with the quiet hum of lawmakers and lobbyists making their daily rounds.

Faced with devastating budget cuts from Gov. Andrew M. Cuomo and a deeply hostile electorate, New York’s most influential public-employee unions have unexpectedly shifted their strategy for defending cherished government programs and worker benefits. Put off for now are the angry denunciations and millions of dollars of advertisements, chiefly from hospitals and a health care union, that have traditionally begun haunting governors in early February.

Instead, two coalitions of labor unions and their allies are mounting campaigns aimed chiefly at persuading Mr. Cuomo to extend the so-called millionaire’s tax, a state surcharge on high-income New Yorkers that is scheduled to expire in December. The tax, which Mr. Cuomo wants to eliminate, could bring in billions of dollars in revenue in the next few years, cushioning cuts to schools and Medicaid.

The labor groups want to reframe a debate that they believe has overemphasized service cuts and cast public workers as selfish and overpaid.

One such group, the Strong Economy for All Coalition, will formally open next week with a $5 million budget, backed by S.E.I.U. 1199, the powerful health care workers’ union, and the United Federation of Teachers, the New York City teachers union.

Most of the money will go not to television and radio advertisements, but for canvassing, social media and other organizing efforts intended to bring pressure on lawmakers from their own constituents, drawing in part on lessons the teachers learned from defeating candidates backed by well-financed charter school advocates in the Democratic primary last fall.

“We think the ad wars make people feel disenfranchised from the process,” said Michael Mulgrew, president of the United Federation of Teachers. “We want to do the grass-roots work — training people on speaking to community groups, going to civic associations, block associations, lawmakers. We want the lawmakers to hear from their real constituents, as opposed to the people that usually lobby them.”

Indeed, an early radio ad script from the group does not mention Mr. Cuomo at all. Instead, it assails “bankers, brokers and money managers” on Wall Street for collecting record pay and bonuses in the midst of a state fiscal crisis.

Such a message contrasts starkly with past budget-battle advertisements. One memorable advertisement run by 1199 two years ago to protest Medicaid cuts featured a blind Bronx resident asking of former Gov. David A. Paterson, who is himself legally blind, “Why are you doing this to me?”

A second coalition, called Growing Together New York, and joining dozens of labor, environmental and community groups, will focus more directly on opposing Mr. Cuomo’s cuts, while also agitating for the extension of the income-tax surcharge. The coalition will be spearheaded by New Yorkers for Fiscal Fairness, a liberal advocacy organization whose backers include CSEA, formerly the Civil Service Employees Association, the largest union of state workers, and New York State United Teachers, the statewide teachers’ union.

A press release announcing the coalition’s formation last week emphasized that it was “not looking for a confrontation with the governor” but instead “a healthy exchange of ideas.”

“We’re mobilizing, and we have a ground strategy,” said Ron Deutsch, executive director of New Yorkers for Fiscal Fairness. “We’re taking the show out on the road, much as the governor has with his budget.”

Mr. Deutsch added, “We want to engage the governor in a dialogue here, and I don’t know if an ad war is the way to do that.”

Both coalitions argue that the debate over New York’s budget crisis has unfairly scapegoated public workers while virtually ignoring the role that Wall Street risk-taking, and the accompanying financial collapse, played. They also say that what Mr. Cuomo has described as an epidemic of overspending is as much a revenue crisis driven by tax cuts for the wealthy from the Pataki era.

The shift in tactics reflects, to some extent, Mr. Cuomo’s efforts to both woo and fracture his opponents, hoping to avoid the kind of joint assault from teachers, health workers and state employees’ unions that ultimately wore down Mr. Paterson and past governors.

Last week, Mr. Cuomo proposed major cuts to Medicaid, school aid and the state work force, while inviting unions to come to him with their own proposals to save money. At the same time, he has amassed a $5 million war chest of his own, money Mr. Cuomo has said he will spend on advertisements only if he is attacked first. He has helped organize a coalition of business groups, called the Committee to Save New York, with plans to spend at least $10 million this year defending his proposals to cap property taxes and rein in spending.

Mr. Cuomo also struck a deal last summer with the Working Families Party, long a potent attack force for unions on fiscal issues, to appear on their ballot line in exchange for their pledge to support his agenda, including his promise not to raise or impose new taxes. The deal did not prevent them, union officials said, from advocating for an extension of the surcharge, an existing tax.

“We are working right now with the governor’s folks on the Medicaid redesign team, and we’re hoping to come up with some creative ways to save money,” said Kevin Finnegan, the political director for 1199.

Some union officials and their allies say the change in strategy also reflects a recognition that the traditional winter ad wars are not without cost to themselves, both in dollars — 1199 alone has spent as much as $10 million a year out of a joint fund with the Greater New York Hospital Association — and in public relations backlash.

“We have had enormous experience with the old way,” said Kenneth E. Raske, president of the hospital association. “The old way leaves a sour taste in our mouth, too. We’ve seen what happens in the past. We didn’t like the process and we didn’t like the outcome. You get some restorations, but you still end up with some difficult cuts.”