Some of the biggest players on the right are involved. The Koch network, the National Rifle Association, the U.S. Chamber of Commerce, and various socially conservative and faith-based organizations are using the campaign as a dry run for untangling the conservative crosscurrents they will have to manage with Mr. Trump as the party’s leader.
Ms. Handel is struggling with the same knot. She welcomed the president for a fund-raiser last month, but kept it closed to the public. She batted away reports that Mr. Trump had shared highly classified information with the Russians by saying they could just be a “gross assumption” by the media, a view many in the district share. But she added that she supported an investigation to resolve the matter.
And when the White House sends in reinforcements to campaign with Ms. Handel ahead of Election Day on June 20, Vice President Mike Pence will be the headliner, not Mr. Trump.
The trick for Republicans and their allied outside groups is figuring out how to avoid conspicuously embracing the president without alienating conservative voters who would view any overt rebuff as a betrayal.
“That is the question we are trying to answer right now,” said Ralph Reed, whose Faith and Freedom Coalition is based in Georgia and is involved in the special election.
As for Mr. Trump, Mr. Reed added: “I don’t think you really look to broadcast him. You narrowcast him.”
The Faith and Freedom Coalition’s “narrowcast” comes in the form of about 30,000 one-page leaflets that volunteers will slip into church bulletins across the district ahead of the election. They list a series of issues that are resonant with conservatives and at the top of Mr. Trump’s agenda — like Planned Parenthood funding, “abortion on demand,” freedom of speech for churches and “amnesty for illegal immigrants” — and offer a side-by-side comparison of Ms. Handel’s and Mr. Ossoff’s stands.
The coalition has identified more than 100,000 potential voters it will target in 250,000 phone calls, 22,000 home visits and 12 videos that will be sent to their phones and computers.
At the other end of the spectrum are groups like the Chamber of Commerce, Americans for Prosperity and the Congressional Leadership Fund, which are determined to keep their distance from Mr. Trump. The Congressional Leadership Fund plans to have at least 20 offices up and running by the end of the year in districts across the country, including one in Roswell that will stay open after the June election.
Those offices will be primarily in districts that are a lot like Georgia’s Sixth: suburban centers populated with highly educated, well-off voters who are not terribly enthusiastic about Mr. Trump. Those districts, in places like California, New Jersey and Florida, will be the terrain on which the Republican House majority is won or lost.
Ms. Handel, a former Georgia secretary of state, embodies the contradictions and competing interests that will make races like hers so hard for Republicans. As a more conventional chamber of commerce-type Republican, she appeals to the party’s traditional constituency. But she is also struggling to excite the voters who rejected that type of Republicanism when they voted for Mr. Trump last year.
“I’m a little nervous,” admitted John Rogers, a real estate broker who answered the door of his home in Roswell recently when a high school volunteer for the Congressional Leadership Fund paid a visit. Mr. Rogers, who describes himself as a conservative but not a Republican, is dismayed by Mr. Trump’s first few months in office.
“The things that come out of his mouth just blow my mind,” he said.
He worried that Democrats were drawing a lot of motivation from the president’s scandals, energy that he said he did not see on the Republican side. “The Democrats at this point are so riled up. And they have a lot to prove,” he said.
The message that groups like the Chamber of Commerce and the Congressional Leadership Fund are pushing in Georgia is Trump-agnostic. The goal is to remind conservative-leaning voters of the common enemy they have in Washington Democrats, even if the headlines about Mr. Trump are too ubiquitous to avoid.
This has meant portraying Mr. Ossoff, a 30-year-old filmmaker who had never run for office before and lives outside the district, as a pawn of Nancy Pelosi, the House Democratic leader, and someone who has been bought and paid for by donors in San Francisco and New York.
“He’s supported by foreigners,” said Julia Bridges, a retired college secretary who used air quotes to indicate that she did not mean actual foreigners but “people from California and places like that.” As she took a break from calling voters from the Faith and Freedom Coalition offices, she added disapprovingly, “We don’t much care for that.”
A new commercial from the Congressional Leadership Fund, which plans to spend $7 million on the race and knock on the doors of 300,000 homes before the election, depicts people on the streets of San Francisco snidely thanking Georgians for Mr. Ossoff. “We already have Nancy Pelosi as our congresswoman,” says one actor in Willie Nelson-style braids as he flashes a peace symbol to the camera. “Now you’re going to give us Jon Ossoff as our congressman.”
But the president does not exactly possess the retiring persona that could make diverting the focus away from him easier.
“This race is a lot closer than it should be, and that’s for one reason: Donald Trump,” said Whit Ayres, a pollster who is advising Republicans on the Georgia race.
If the debate is primarily about the president, Republicans cannot easily shift the focus toward more parochial matters — like money for public works, local military bases and schools — that would usually allow them a measure of political isolation.
“What’s driving this race,” Mr. Ayres added, “are broader national factors that are not in anyone’s control.”
The national implications seem clear to many voters, who understand the vulnerability that a Republican loss would telegraph to the country. Bob Harris, a retired sixth-grade science teacher who lives in once reliably Republican east Cobb County, said he was not exactly “on fire when it comes to Karen Handel.” He wishes Mr. Trump would get off Twitter and take a lesson in composure from more coolheaded advisers like Rex W. Tillerson, the secretary of state, and Reince Priebus, the White House chief of staff.
But he wants a Republican to win the seat so badly that he is volunteering for Americans for Prosperity, the group funded by the billionaire conservatives Charles G. and David H. Koch, and was making phone calls to voters from its office in Marietta on a recent rainy evening.
“I wouldn’t bet any money on this one,” he said, acknowledging how close the race seemed. “The Democrats smell blood in the water.”