With the Democratic primary at an inflection point as Mr. Sanders considers his next steps, Mr. Biden’s campaign has tried to strike a balance between acting presidential, as if he will be the nominee, and not alienating Mr. Sanders and his loyalists. Last Wednesday, hours after Mr. Biden had collected four more victories, his aides seemed reluctant to call on Mr. Sanders to drop out even as they sent out a memo describing Mr. Biden’s lead as nearly insurmountable.

“Should our broad base of support remain — and we have seen no signs that would indicate otherwise — it will be nearly impossible for Sanders to recoup his current delegate disadvantage,” the Biden campaign memo said.

After Mr. Sanders refrained from attacking Mr. Biden at a news conference that afternoon, and instead presented a list of policy priorities, some of Mr. Biden’s backers hoped the debate would be the next step toward unifying the party around his candidacy. Mr. Biden’s team believed it had extended an olive branch to Sanders loyalists in the hours before the debate, embracing a version of a plan Mr. Sanders has championed that calls for tuition-free public college for many students.

So some Biden backers said they were caught off guard when Mr. Sanders spent much of Sunday’s debate criticizing Mr. Biden’s long record, which revived their broader frustrations with some of Mr. Sanders’s surrogates who have slashed Mr. Biden in more starkly personal terms.

“I didn’t expect some of the sharp elbows we saw from the senator,” said state Representative Malcolm Kenyatta, Democrat of Pennsylvania and a Biden supporter, even as he praised Mr. Sanders’s pledge to support the eventual nominee.

There were more overt indications of displeasure from the Biden camp directly after the debate. Anita Dunn, Mr. Biden’s chief strategist, compared Mr. Sanders to a political demonstrator, telling reporters that the former vice president had spent the debate “graciously dealing with the kind of protester who often shows up at campaign events, on live television.”

Some Sanders advisers are still hoping to eke out victories on Tuesday in Illinois and Arizona but are also cognizant that the outlook looks increasingly challenging. A poll released on Monday showed Mr. Sanders trailing Mr. Biden in Arizona by 20 percentage points. Florida also votes on Tuesday, and aides acknowledge that the state at this point is effectively out of reach. And in a blow that was long in coming, Washington State’s final results were announced on Monday, with Mr. Biden emerging as the victor after nearly a week of counting ballots.