Mr. Sanders and Ms. Warren have accused one another of lying about a private conversation in 2018 over whether a woman could become president; Mr. Sanders and Mr. Biden have attacked each other over Social Security and corruption; and Hillary Clinton, the 2016 Democratic nominee, has come off the sidelines to stoke her rivalry with Mr. Sanders, declaring that “nobody likes him.”

The lack of consensus among Democratic voters, 10 days before the presidential nominating primary begins with the Iowa caucuses, has led some party leaders to make unusually fervent and early pleas for unity. On Monday alone, a pair of influential Democratic congressmen issued strikingly similar warnings to very different audiences in very different states.

“We get down to November, there’s only going to be one nominee,” Representative James E. Clyburn of South Carolina, the third-ranking House Democrat, said at a ceremony for Martin Luther King’s Birthday at the State House in Columbia. “Nobody can afford to get so angry because your first choice did not win. If you stay home in November, you are going to get Trump back.”

“No matter who our nominee is, we can’t make the mistake that we made in ’16,” Representative Dave Loebsack of Iowa said that night in Cedar Rapids as he introduced his preferred 2020 candidate, former Mayor Pete Buttigieg of South Bend, Ind., at a town hall meeting. “We all got to get behind that person so we can get Donald Trump out of office,” Mr. Loebsack added.

In interviews, Democratic leaders say they believe the party’s fights over such politically fraught issues as treasured entitlement programs, personal integrity, and gender and electability could hand Mr. Trump and foreign actors ammunition with which to depress turnout for their standard-bearer.

“I am concerned about facing another disinformation campaign from the other side,” said Representative Brendan Boyle of Pennsylvania, a Biden supporter who was uneasy enough that he recently sought out high-profile congressional backers of some of the other contenders to discuss an eventual détente. “For those of us who are elected officials, we need to exercise real leadership to make sure all of the camps are immediately united after all this is over.”