“Our brand is worse than Trump,” said Mr. Ryan, who urged Democrats to make forging a clear economic agenda an urgent priority. “We can’t just run against Trump.”
Mr. Ryan, who tried to unseat Ms. Pelosi, Democrat of California, as House minority leader after the November elections, said she remained a political drag on other Democrats. Ms. Handel and Republican outside groups tied Mr. Ossoff to Ms. Pelosi in campaign events and television ads, casting him as a puppet for what they described as her liberal agenda and “San Francisco values.”
“They’re still running against her and still winning races, and it’s still a problem,” Mr. Ryan said.
In Washington, Representative Ben Ray Luján of New Mexico, the chairman of the Democratic Congressional Campaign Committee, moved to calm the party overnight, circulating a memo that outlined in detail how Democrats aim to capture a majority in 2018. In the document, which was sent to lawmakers and staff, Mr. Luján wrote that there was “no doubt that Democrats can take back the House next fall” in the midterm elections.
Acknowledging that the Georgia result was a setback, Mr. Luján wrote on Wednesday that there were between six and eight dozen seats held by Republican lawmakers that would be easier for Democrats to capture than Georgia’s Sixth. He said the next few months would become a “recruitment blitz” for Democrats as they enlist candidates in those elections.
“Let’s look outside of the traditional mold to keep recruiting local leaders, veterans, business owners, women, job creators and health professionals,” Mr. Luján wrote. “Let’s take the time to find people who fit their districts, have compelling stories, and work hard to earn support from voters.”
And, citing snippets of private polling, Mr. Luján said there were Republican seats in southern Arizona and Florida, northern New Jersey and the Kansas City, Kan., suburbs, where Democratic challengers were already ahead of Republican incumbents.
Democrats need to win 24 Republican-held seats in order to win control of the House.
Democratic lawmakers are expected to gather Wednesday morning in a caucus meeting, which could become a forum for restive lawmakers to air their political anxieties and grievances.
But well before the gathering, a half-dozen Democratic elected officials and operatives privately vented in text messages and phone calls about a dispiriting trend emerging in this year’s special elections: When their candidates appear to gain traction in the polls, Republicans can easily halt the momentum by invoking Ms. Pelosi.
A spokesman for Ms. Pelosi noted that in some polls House Speaker Paul D. Ryan’s approval ratings were even more dismal than Ms. Pelosi’s and argued that the right would make any high-profile Democratic leader the focal point of attacks.
“Republicans blew through millions to keep a ruby red seat and in their desperate rush to keep stop the hemorrhaging, they’ve returned to demonizing the party’s strongest fund-raiser and consensus builder,” said Drew Hammill, Ms. Pelosi’s deputy chief of staff. “They don’t have Clinton or Obama so this is what they do.”
On the Republican side, jubilation over Ms. Handel’s victory mixed with lingering unease about the overall political environment. While Ms. Handel defeated Mr. Ossoff by about 10,000 votes and nearly four percentage points, Republican outside groups had to spend $18 million defending a district where the party’s candidates won easily for decades.
Mr. Trump’s approval ratings continued to fall steadily over the course of the race, and Ms. Handel at times took active steps to distance herself from the president.
And on the same night, a little-watched special election in South Carolina gave the Republican Party another scare, as an obscure Democrat, Archie Parnell, came within 3,000 votes of capturing a solidly Republican congressional district. Turnout in the election was tepid, and the close margin was seen in both parties as a function of Democrats’ greater enthusiasm for voting in a low-key race.
Nick Everhart, a Republican strategist in Ohio, said the party should not allow its relief at having kept Democrats at bay turn into complacency. Up to this point, he said, Republicans have been beating Democrats only on solidly red turf.
“To pretend that there are not serious enthusiasm-gap issues with the G.O.P. base and more crucially, independents fleeing, is missing the lessons that need to be learned before truly competitive seats are on the board,” Mr. Everhart said.
Still, the immediate aftermath of the Georgia election was plainly tougher on the Democratic side, where activists, donors and lawmakers have grown weary of special elections that end with a better-than-usual showing by a defeated Democrat. That pattern, which also characterized recent congressional races in Kansas and Montana, may put Democrats on track to gain power in the 2018 elections, but 17 months is a long wait for a party hungry for immediate victories.
The lack of a signal success for Democrats during this season of special elections has also left open existential questions for the party, which might have been resolved by a smashing win of one kind or another. Had Mr. Ossoff won on Tuesday night, it most likely would have emboldened the party’s moderate establishment and made his campaign — waged in broad promises of economic growth and cautious language about Mr. Trump — a model for other races.
Instead, populist forces on the left took Mr. Ossoff’s defeat as an occasion to criticize the whole notion of centrism as a Democratic strategy. Jim Dean, the chairman of Democracy for America, a liberal activist group, blasted Mr. Ossoff overnight for “lighting millions of dollars on fire” and delivering an “uninspiring message” that he predicted would fail again in 2018.
“The same, tired centrist Democratic playbook that has come up short cycle after cycle will not suffice,” Mr. Dean said in a statement.
Outside the activist wing of the party, the Georgia result opened a new and potentially bitter debate about nonideological matters, including who should be the Democrats’ public face in Congress.
Lachlan McIntosh, a South Carolina-based Democratic strategist, said the party had to confront just how unpopular its current leadership is in conservative areas. With some Democrats questioning whether they should have invested more in Mr. Parnell’s near-miss campaign, Mr. McIntosh said bluntly that, if Mr. Parnell had ever become a cause célèbre like Mr. Ossoff, national Republicans would simply have blasted him with the same anti-Pelosi message that worked in Georgia.
“The problem will persist for Democrats next year,” Mr. McIntosh said. “They can’t ignore how very unpopular their leadership is with many voters. It’s not fair or justified, but it’s real.”