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.