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An overwhelming majority of Republican voters say their party’s leaders should get behind Donald J. Trump, even as he enters the general election saddled with toxic favorability ratings among the broader electorate, according to the latest New York Times/CBS News poll.

And as Mr. Trump faces deep skepticism with general election voters and some Republican holdouts, the Democratic front-runner, Hillary Clinton, is grappling with Senator Bernie Sanders and how to win overhis impassioned supporters.

Both parties thus approach their July nominating conventions with significant unease and hurdles to overcome. Mr. Trump and Mrs. Clinton are widely disliked by voters, and both parties will need to repair schisms that might spell doom in an ordinary election year. But this, of course, is no ordinary year.

Mr. Trump’s and Mrs. Clinton’s soaring levels of unpopularity are extraordinary for the likely nominees of the two major parties. Nearly two-thirds of voters , for example, say that Mr. Trump is not honest and trustworthy. Just as many say the same of Mrs. Clinton. Strong majorities of voters say the candidates do not share their values.

Mr. Trump’s difficulties appear to be more troublesome at the moment. If the election were held now, 47 percent of registered voters would support Mrs. Clinton, versus 41 percent for Mr. Trump. Mrs. Clinton’s head-to-head advantage has narrowed somewhat since Mr. Trumpbecame the presumptive Republican nominee: Last month, she led him by 10 points in a CBS News poll.

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Hillary Clinton and Bernie Sanders.CreditLeft: Ty Wright for The New York Times; Right: Eric Thayer for The New York Times

In a more hypothetical matchup, Mr. Sanders leads Mr. Trump, 51 percent to 38 percent.

The survey reveals that Republican voters are starting to fall in line with Mr. Trump now that he is their presumptive nominee — and that they expect party officials to do the same.

Eight in 10 Republican voters said their leaders should support Mr. Trump even if they disagree with him on important issues. And unfavorable views toward Mr. Trump among Republican voters have plummeted 15 percentage points since last month; 21 percent now express an unfavorable view of him, down from 36 percent in April.

Republican voters are drifting toward unity even though some party elites are still withholding their support. The House speaker, Paul D. Ryan, continues to deliberate over whether to endorse Mr. Trump, while other prominent Republicans are actively searching for a third-party nominee as an alternative.

Yet many of the party’s rank-and-file voters think that while Mr. Trump may be imperfect, the time has come to rally to their unlikely standard-bearer — if only to keep Mrs. Clinton out of the White House.

“The reason I would support him is because the alternative is less favorable, in my opinion,” Delores Stockett, 76, a retired teacher from Osceola, Ark., said in a follow-up interview. “And I think Republican leaders should support him for the same reason I am supporting him. The alternative is not palatable to those of us who hold conservative views.”

Are You Concerned About the Republican Party’s Future?

There is deep concern among Republicans about political divisions and the future of the party, a new Times/CBS News poll shows. We want to hear from the party’s longtime members or those who have just registered as Republicans.

Republican voters remain remarkably pessimistic about the state of their party. More than eight in 10 call the party divided, and 43 percent say they are discouraged about its future. About two-thirds think Mr. Trump can unite the party this year.

By contrast, fewer than half of Democratic voters say their party is divided, and eight in 10 are hopeful about its future. More than eight in 10 think Mrs. Clinton can unite the party after the primaries end next month.