When Senator Mike Rounds, Republican of South Dakota, first saw President Trump’s request to cancel $9 billion in congressionally approved funding for foreign aid and public broadcasting, he balked.

He was wary of supporting a measure that would gut public media, given that his state relies so heavily on federally funded tribal stations. But before long, top Trump administration officials were assuring him that they would find grants to fund those outlets. That vague assurance was enough to win over Mr. Rounds, who ultimately voted for the measure.

It was part of a pattern that has emerged as Republicans in Congress, increasingly ceding their power to Mr. Trump, continue to find ways of falling in line behind even elements of his agenda that run directly counter to their interests and stated priorities.

Republican leaders presiding over slim majorities in both chambers are more and more dependent on the president and his team to cut side deals with holdouts to win enough votes to push through his top priorities.

That has been the case for some of Mr. Trump’s biggest legislative items, including the massive tax cut bill that slashed Medicaid and food assistance, and a bill clawing back $9 billion in foreign aid and public broadcasting funds that Congress had already approved. In both cases and many others, skeptical G.O.P. lawmakers have relented after winning promises — sometimes little more than imprecise commitments for future action — that their pet issues will be addressed or their constituents spared the worst of the impacts.

Horse-trading behind the scenes has always powered much of what goes on in Congress, particularly when it comes to major legislative pushes. But the past several weeks have been dominated by an extraordinary flurry of handshake deals and written carve-outs meant to assuage the most anxious Republicans, often with scant details about how the exceptions will be implemented or enforced.