ALEXANDRIA, Va. — During one of the most hectic weeks of her speakership — as she sought to unite her fractious party and corral two sweeping pieces of legislation — Nancy Pelosi made time for a meeting in her Capitol suite with a group of Democratic lawmakers from New Jersey and Virginia bearing an urgent message of their own.

They warned Ms. Pelosi that if the candidates for governor in those two states, particularly former Gov. Terry McAuliffe in liberal-leaning Virginia, were to lose on Tuesday, it could have a cascading effect on the party, prompting Democrats to pull back from President Biden and his ambitious agenda, and perhaps even drive some to retirement.

Representative Gerald Connolly of Virginia said he used the meeting last Tuesday to urge Ms. Pelosi to pass the bipartisan infrastructure bill, which had already cleared the Senate, and to share his alarm about the party’s fortunes. “You don’t have to be a front-liner to be worried,” he said, invoking the word House Democrats use to describe their most politically at-risk incumbents.

Unable to overcome mutual mistrust between a group of House progressives and Senate moderates, however, Ms. Pelosi pulled the public works legislation from consideration hours after Mr. Biden visited the Capitol on Thursday, dashing the group’s hopes of delivering Mr. McAuliffe and other Democrats on the ballot a win after a two-month drumbeat of bad news.

The former Virginia governor and his top aides, who have been pushing congressional and White House officials to pass the bill for over a month, were both stunned and infuriated, according to Democrats. They were amazed Ms. Pelosi had been forced to delay the vote for the second time in a month, baffled why the president didn’t make a more aggressive push and despairing about the impact of yet another round of negative stories from Washington.

“The last two-and-a-half months makes it look like Democrats are in disarray,” said Representative Filemon Vela, a Texas Democrat who has raised money for Mr. McAuliffe.

The races for governor in Virginia and New Jersey that occur a year after the presidential election have long been the first political temperature checks on the new White House and Congress, particularly among the election-deciding suburbanites so abundant in both states. But rarely have contests traditionally fought over decidedly local issues been so interwoven with the national political debate and, in the case of Virginia, loomed as so large a portent for the future of both parties.

Mr. McAuliffe’s strategy of relentlessly linking his Republican rival, Glenn Youngkin, to Donald J. Trump represents the best test yet of how much of a drag the former president still exerts on his party in blue and purple states. At the same time, Mr. Youngkin’s fancy footwork regarding Mr. Trump — avoiding his embrace without alienating him or his base — and his attacks on Mr. McAuliffe over the role of parents in schools will indicate if G.O.P. candidates can sidestep Trumpism by drawing attention to what they argue is Democratic extremism on issues of race and gender.

Far from such old standbys of statewide races as property taxes and teacher pay, the issues in Virginia reflect the country’s canyonlike polarization and what each party portrays as the dire threat posed by the other.

To Republicans, Virginia represents the promise of renewal, the chance to rebuild their party in a fairly forbidding state — and without having to make the difficult choice of fully embracing or rejecting Mr. Trump. Addressing supporters near a farmer’s market in Old Town Alexandria Saturday morning, Mr. Youngkin said his victory would send “a shock wave across this country.”

Suffering yet another loss here, though, would make it clear to Republicans that they cannot continue to delay their internal reckoning over the former president and that, even in exile, his unpopularity remains the party’s biggest impediment.

Because Mr. Biden carried the state by 10 points last year, and Mr. McAuliffe began the race with an advantage befitting the former governor he is, the most significant implications in Virginia are for Democrats. The party is also haunted by recent history: Their loss in the 2009 Virginia governor’s race — the last time Democrats controlled the White House and both chambers of Congress — foreshadowed the party’s electoral wipeout the following year.

Should Mr. McAuliffe lose or barely win, moderates will demand immediate passage of the infrastructure bill. Liberals will argue that the Democratic Party, and democracy itself, are in such a parlous state that they must push through new voting laws. And strategists across the party’s ideological spectrum will be made to contend with a political playing field in the midterm elections that stretches deeper into blue America.

“We’re going to have to change our calculation of what’s a race and look at the districts Trump lost,” said Rebecca Pearcey, a Democratic consultant. “Even if he wins,” she added, referring to Mr. McAuliffe, “we’re going to have to reassess what the map looks like on Wednesday, because Tuesday is not going to be a pretty night.”

With Mr. Biden’s approval ratings tumbling among independents thanks to Covid-19’s summer resurgence, the botched Afghanistan withdrawal and rising inflation, Democrats are also bracing for additional retirements among lawmakers who would rather not run in a newly redrawn district or risk ending their careers in defeat.

Already, three House Democrats announced their departures earlier this month. Mr. McAuliffe’s defeat in a state Mr. Biden so easily won would likely accelerate that exodus because lawmakers will chalk it up to the president’s unpopularity. “We’re going to see a lot more by the end of the year,” predicted Ms. Pearcey.

Ms. Pelosi is acutely aware of these flight risks — she herself is one — and has privately expressed concern about the fallout from Mr. McAuliffe’s race. Last week, she told a Democratic colleague that the House’s failure to pass the infrastructure bill could imperil Mr. McAuliffe, according to a lawmaker familiar with the exchange.

A longtime friend of Mr. McAuliffe’s, Ms. Pelosi headlined a fund-raiser for him last week and has personally given him $250,000 and raised over three times as much. She has also spoken with him by telephone repeatedly about negotiations over the infrastructure bill.

A backlash next year may be inevitable in part because, as one longtime Democratic lawmaker noted, the party often suffers at the polls after it pushes an expansive agenda of the sort that congressional Democrats are painstakingly negotiating.

“Somebody reminded me: In ’66, after all we did in ’65, we got beat,” Representative Robert C. Scott of Virginia said, referring to the losses Democrats incurred after passing much of the Great Society. “We passed Obamacare and we got beat.”

Much as the protracted debate over the Affordable Care Act in 2009 and 2010 overshadowed their economic-recovery legislation then, Democrats this year have been more focused on negotiating their towering twin bills than on promoting their earlier Covid relief legislation.

While Democratic lawmakers have dwelled almost exclusively on the infrastructure bill and their broader social welfare and climate proposal — matters on which they have not reached consensus — Americans outside of Washington have grown impatient with the lingering virus and the soaring prices of goods.

“If you listen to the Democratic speaking points, it’s all what we haven’t done,” said Mr. Scott, pointing out that the child tax credit enacted in the Covid rescue plan earlier this year was often left unmentioned.

Mr. Vela, a moderate who has demanded an infrastructure vote since August, said a McAuliffe defeat should prompt quick passage of that bill, which passed the Senate with 69 votes. “Progressives should wake up and realize that linking the two processes together was a huge mistake,” he said, adding: “That’s from somebody who supports both bills.”

But many on the left believe that the party’s vulnerabilities, laid bare by the prospect of defeat in Virginia, where they have not lost a statewide race since 2009, only underscore the need to scrap the Senate filibuster and push through sweeping voting laws that could stave off a disastrous 2022 and long-term loss of power.

“A close race in Virginia would signal just how hard the midterms will be for Democrats and the urgency of passing democracy reform,” said Waleed Shahid, a spokesman for the left-wing Justice Democrats who grew up in Virginia.

Progressives were already unenthusiastic about Mr. McAuliffe, a fixture of the party establishment and a former national Democratic chairman, and they have been irritated that he did not do more this year to help the party’s state lawmakers hold onto the majority they won in the House of Delegates in 2017.

“Whatever happens Tuesday, one lesson we already know from Virginia is that we better prioritize winning state legislatures like our democracy depends on it — because it does,” said Daniel Squadron, who runs the States Project, which is dedicated to electing Democrats in statehouses.

To Democrats in Northern Virginia, who prospered in the Trump years, there is an especially close connection between what takes place in the nation’s capital and their seats.

Mr. Connolly recalled that at the Halloween parade last week in Vienna, Va., a handful of people yelled at him to pass the infrastructure bill, a major quality-of-life issue in his traffic-choked district. “It really got my attention,” he said.

Now, he said, he hopes it will not take another high-profile loss in his home state to get his party’s attention.

“If past is prologue,” he said, “we cannot have a repeat of what happened in 2009.”