RHINEBECK, N.Y. — Even before Peter Plavchan Sr. asked about “the exact crimes” President Trump had committed to prompt his congressman to reverse course and embrace an impeachment inquiry, Representative Antonio Delgado was ready with a detailed answer.

“We are literally about to embark on something that has not happened all that often in our country’s history,” Mr. Delgado, Democrat of New York, told Mr. Plavchan and other constituents last week in his opening remarks at a town hall-style meeting in his district. Pressed further, his voice began growing hoarse as he explained, at length, the allegations about Mr. Trump’s pressure campaign on Ukraine.

“I’m endeavoring, in this space, to the best of my ability, to do whatever I can to not just fan the flame, but to figure out how to bring us through this process in a way that can very least help us emerge better for it,” Mr. Delgado said.

It is a difficult balance to strike, with Washington in a partisan frenzy over an impeachment inquiry that expands by the day as new revelations about Mr. Trump’s conduct pour out, but much of the nation is still divided over — or simply uninterested in — a process that could result in the removal of the president.