ALPHARETTA, Ga. — A gray mood has settled over conservative-leaning voters in some of the country’s most reliably Republican congressional districts, as the party’s stumbles in Washington demoralize them and leave lawmakers scrambling to energize would-be supporters in a series of off-year elections.

While the next nationwide elections are not until 2018, Republicans have grown fearful that these voters are recoiling from what they see as lamentable conditions in Washington: a government entirely in Republican hands that has failed to deliver on fundamental goals like overhauling the health care system.

Early missteps by President Trump and congressional leaders have weighed heavily on voters from the party’s more affluent wing, anchored in right-of-center suburbs around major cities in the South and Midwest. Never beloved in these precincts, Mr. Trump appears to be struggling to maintain support from certain voters who backed him last year mainly as a way of defeating Hillary Clinton.

Interviews with Republican-leaning voters in four suburban districts — in Georgia, Kansas, Minnesota and New Jersey — revealed a sour outlook on the party. These voters, mainly white professionals, say they expected far more in the way of results by now, given the Republican grip on power in the capital. In opinion polls, they consistently give Mr. Trump mediocre approval ratings, even as he remains solidly popular with lower-income whites.