MANCHESTER, N.H. — Former Vice President Joseph R. Biden Jr.’s fourth-place finish in Iowa and his wobbly standing here in New Hampshire are now testing the central premise of his candidacy — that he is the strongest Democrat to defeat President Trump — and forcing his team to scramble to prove that claim before voters move on to other candidates.

It’s a striking departure from Mr. Biden’s self-assured posture throughout the campaign. He has said he is “the clear front-runner in the party.” His allies still cite his strength in general election polls constantly, and have even featured them in his television commercials. He has spent months targeting Mr. Trump rather than driving a sustained message at his rival Democrats, and his attempts to do so at Friday night’s debate, while aggressive, did not appear to hurt other candidates.

Now, as he prepares for New Hampshire’s primary on Tuesday, Mr. Biden’s campaign is confronting its greatest moment of peril to date, marked by worrisome polls, jittery donors and tensions within the staff.

“If your whole theory of the case is that I’m the electable one and I can win, and then you lose in the first state and possibly the second state, it sort of blows your entire message,” said Patti Solis Doyle, who served as Hillary Clinton’s campaign manager in 2008.

Another weak showing, some Democrats warn, could accelerate a flight of wealthy backers to his two most formidable moderate rivals: Pete Buttigieg and Michael R. Bloomberg. It could also jeopardize the former vice president’s strength in the later-voting, more diverse states he is counting on.

In a sign that he knows his campaign is in trouble, Mr. Biden shook up his leadership team at the end of the week, turning over effective control of the campaign to a longtime adviser, Anita Dunn (his current campaign manager, Greg Schultz, was spotted at a Democratic dinner in Washington instead of at Friday’s debate). News of the shake-up leaked quickly and, in an illustration of the growing angst inside his campaign, a number of aides privately indicated how happy they were about the change, while others sought to downplay it.

But no amount of staff reshuffling may address Mr. Biden’s more fundamental challenge: The setup of the primary calendar means that Mr. Biden is faltering before he has even had a chance to compete in states where he has broader support, like Nevada and South Carolina later in the month.

And without a turnaround, he will not have the money to compete on Super Tuesday on March 3, when 14 states, including California and Texas, go to the polls.

Of the top-four finishers in Iowa, Mr. Biden entered 2020 with the smallest amount of cash on hand — less than $9 million — and his January fund-raising total is expected to be less than half of the $25 million haul that Senator Bernie Sanders of Vermont announced on Thursday. Mr. Biden has been vastly outspent on television ads in New Hampshire, and this past week he shifted some ad spending to Nevada from a state with a later contest, South Carolina.

Being forced out by money woes would be a humbling scenario for a former vice president making his third attempt at the White House, who entered the race with widespread name recognition, deep relationships across the party and the reflected glow of having served in the Obama White House.

“The best way to prove electability is to do well in contests, and this was always a risk for him because he has hung his hat so strongly on the idea that he was the most electable candidate,” said Dan Pfeiffer, who served as a top aide to Barack Obama. Referencing the New Hampshire contest, he said, “The test for Biden is less about where he places and more about whether he shows strength as a candidate coming out of it.”

Troubled by Mr. Biden’s poor showing in Iowa, a number of his top supporters have been urging his campaign to make changes: Have the former vice president do more news media interviews. Spend more time engaged in retail politics. “Let Joe be Joe.”

“He has to get out there more, so people see him and talk to him,” said Mr. Biden’s national campaign co-chairman, Representative Cedric L. Richmond of Louisiana, when asked what Mr. Biden needed to be doing differently. “When he does that, he wins people over.”

Last Wednesday Mr. Biden appeared at a CNN town hall event, and impressed many observers by speaking movingly about helping people who struggle with stuttering — as he once did. Afterward, a prominent New Hampshire Democrat cornered him.

That was great, State Senator Lou D’Allesandro recalled telling him, impressed by the connection with the audience. Do more of that.

“He’s got to get into full-court press mode,” said Mr. D’Allesandro, a Biden supporter. “He’s got to be, every day, like he was at the CNN town hall meeting. If you want to be president, that’s what you’ve got to do.”

Yet Mr. Biden proceeded to go home to Delaware to prepare for Friday night’s debate, taking two days off the New Hampshire trail at one of the most uncertain, high-risk moments of his campaign before returning for the debate and a busy final weekend push.

Asked if he believed Mr. Biden was making that “full-court press,” Mr. D’Allesandro replied: “I hope so. I’m trying to convince them that’s what they should do.’’

While Mr. Biden was out of state, his chief rival for moderate votes in New Hampshire, Mr. Buttigieg, the former mayor of South Bend, Ind., held an event with veterans and drew substantial media coverage.

Several attendees fit the profile Mr. Biden is targeting: older, moderate, affiliated with labor and skeptical of far-reaching proposals like “Medicare for all.” It was a sign of the fierce competition Mr. Biden faces for the centrist voters his allies had hoped he would own — and at least at Mr. Buttigieg’s event, a number of attendees who said they had been torn between the two candidates indicated that they were prepared to settle on the young former mayor.

“I think that’s a factor,” said Steve Shurtleff, the New Hampshire House speaker, who has endorsed Mr. Biden. “I thought it would be an easy decision, where I’d supported Joe Biden in the past, but I really gave a lot of thought to it. I decided to support the vice president, but I was very impressed with Mayor Pete.”

He questioned Mr. Buttigieg’s experience but added, “I’m sure a lot of people are looking at the vice president and Mayor Buttigieg and making that decision.”

Mr. Biden’s campaign has been adamant that when they discuss electability, they are focused on the general election against Mr. Trump, arguing that the former vice president has the unique ability to both assemble a diverse coalition and to compete in the industrial Midwest. Mr. Buttigieg, Mr. Biden himself has suggested, struggles with voters of color, a bedrock of the Democratic Party; he did not register with black voters in a recent Quinnipiac University poll.

“Joe Biden is a fighter,” said Andrew Bates, a spokesman for Mr. Biden’s campaign. “And he’s going to fight for every last delegate throughout this contest as millions of Democratic voters weigh in on who can actually go toe-to-toe with Donald Trump, win in key battleground states, take the Senate and turn progressive plans into reality.”

Mr. Biden is also supported by a super PAC, and his campaign said that Friday was his best debate day for online fund-raising, but did not announce a number.

How Mr. Biden fares against his Democratic rivals in a nominating contest in a single state — especially in the heavily white states of Iowa and New Hampshire — is not necessarily predictive of how he would do in later contests, or against Mr. Trump in battleground states in a general election. But his loss in Iowa was not a narrow one. Of the 31 counties in Iowa that voted for Mr. Obama in the 2008 and 2012 elections only to flip to Mr. Trump in 2016, Mr. Biden won only one.

Mr. Biden currently has a commanding lead in South Carolina with older African-American voters, a crucial part of the electorate there. But as Tom Steyer demonstrated with his repeated appeals at Friday’s debate, others in the race are aggressively seeking to make inroads with Mr. Biden’s base, a task that could become easier if Mr. Biden finishes down the pack in the first three states.

“I still believe it’s Biden’s race to lose,” said former Gov. Jim Hodges of South Carolina, who has endorsed Mr. Biden. “Nothing in Iowa changed my mind on that. Very little in New Hampshire can change my mind. Now you know, if he performs poorly in New Hampshire and Nevada, that’s not good, but I don’t anticipate that’s the case.”

Campaigning in New Hampshire on Wednesday, Mr. Biden seemed to understand it was time to make his argument more aggressively, and he took explicit aim at the two top finishers in Iowa, Mr. Buttigieg and Mr. Sanders.

At an event in Somersworth where teleprompters had been set up, Mr. Biden argued that putting Mr. Sanders at the top of the ticket would expose down-ballot Democratic candidates to being labeled “socialist,” given Mr. Sanders’s description of himself as a democratic socialist. And he highlighted Mr. Buttigieg’s relative lack of experience, in addition to faulting him for not appropriately crediting the accomplishments of what Mr. Biden likes to call the “Obama-Biden administration.”

It was exactly the kind of feisty performance some of his supporters had hoped to see.

“Voters have to know the difference,” Mr. Richmond said.

Jim Demers, a New Hampshire lobbyist who was an early supporter of Mr. Obama’s, said Mr. Biden was right to be more assertive in contrasting himself with Mr. Sanders and Mr. Buttigieg. Mr. Demers supports Mr. Biden after originally backing Senator Cory Booker of New Jersey.

“People here want to see Joe Biden up for the fight,” he said, “so they’re looking for that fire that they know he has.”

Jonathan Martin contributed reporting from Manchester, and Stephanie Saul from Columbia, S.C.

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