Even as Democrats notched their wins on Tuesday night, more gloomy polls came in illustrating Mr. Biden’s challenges. A new survey by The Atlanta Journal-Constitution and the University of Georgia showed the president in a dead heat with Mr. Trump in Georgia, a state that was critical to Mr. Biden’s victory in 2020.

And for the first time in his presidency, a majority of voters in reliably blue California disapprove of Mr. Biden’s job performance, according to another poll by The Los Angeles Times and the Institute of Governmental Studies at the University of California, Berkeley. Mr. Biden still leads Mr. Trump, and no Democrat is seriously worried about losing California. But the findings underscore the problems of the president, who is 80 years old, with his own base as many Democrats tell pollsters that they think he is too old to run again.

As much as they are frustrated by such polls, some Biden advisers see a silver lining in them because they remind the president’s supporters that next year’s election could be achingly close, thereby motivating them to work harder for his campaign and turn out in greater numbers.

The president’s allies increasingly have come to the conclusion that they may not be able to sell Mr. Biden and instead need to focus more on scaring voters about the possible return of Mr. Trump. In the meantime, they will savor Tuesday night’s elections and use them to bolster Democratic morale.

“The immediate benefits to Biden are mostly narrative,” said Sarah Longwell, a Republican strategist and a leading opponent of Mr. Trump who supported Mr. Biden in 2020. “If you’re his team, you can wave last night’s results in naysayers’ faces and say, ‘See, Democrats keep winning elections under Biden. You keep underestimating him, but he wins when it matters.’ It also disrupts his doomsday news cycle after the bad polls.”

“But ultimately,” she added, “I don’t think last night tells us anything about 2024.” While the New York Times poll showed Mr. Biden’s vulnerability, she said next year’s election “is going to be about Donald Trump.” The task for the Biden campaign then, she said, will be making sure that “it galvanizes the anti-Trump vote, which is a different thing than a pro-Biden vote.”

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