Matt Walter, president of the Republican State Leadership Committee, the party’s national hub for legislative campaigns, said Republicans were on the defensive in all but a few states. Citing Democratic turnout in recent special elections, Mr. Walter said Republicans should use the next nine months to sound the “alarm bells” for their voters.

“What we have seen in the special elections is a significant spike in the interest, engagement, spending and energy by the liberal Democrats and progressive movement,” Mr. Walter said, adding: “The spending is real. The organizational prowess is real. And the energy is real.”

That energy was on raucous display last weekend in the Bucks County borough of Newtown, where well over 100 Democrats packed into a red-brick tavern to cheer Steve Santarsiero, a Democrat seeking a State Senate seat left open by a Republican’s unexpected retirement. Before a lively breakfast crowd, Mr. Santarsiero needled Mr. Trump and hailed his fellow Democrats running for the legislature’s multiplying number of open seats.

Applauding from the front was Helen Tai, an official in nearby Solebury who is running in a May special election for the State House prompted by a Republican’s resignation. Democrats nearly swept local elections in four counties outside Philadelphia last November; Ms. Tai said the combination of Republican retirements and liberal enthusiasm had transformed the fight for the legislature.

“I wish it was a presidential year,” she said. “People want to vote. They can’t wait to vote.”

Adding to Republicans’ unease are several unresolved lawsuits that could unravel carefully drawn maps in states like North Carolina and Texas. The United States Supreme Court is expected to consider a number of cases involving gerrymandered maps this year, and Jessica Post, executive director of the Democratic Legislative Campaign Committee, said the group is considering new litigation against state legislative districts in the Pennsylvania courts, which voided a Republican-drawn congressional map last month.

Ms. Post said special elections over the last year had revealed “early indicators of the wave.”

In many of the biggest purple states, however, Democrats must overcome huge Republican majorities and forbidding legislative maps. In Pennsylvania, Republicans hold 120 seats in the 203-seat State House, and 34 of 50 in the State Senate.