PHILADELPHIA — When President Obama takes the stage here this week to begin his monthslong political farewell, it will mark a bittersweet moment for Democrats.

Mr. Obama is not going to be missed only because he is beloved by the faithful. The president has also fulfilled two longstanding if often elusive goals of his party: He has largely transcended the traditional moderate-versus-liberal split, papering over internal divisions. And he passed a sweeping health care law.

But without the unifying presence of Mr. Obama in the White House, or the organizing aim of expanding health care, it is unclear who will hold the party together and what will be the next great project of liberalism.

For the moment, Hillary Clinton and her party have an industrial-strength adhesive: keeping Donald J. Trump out of the Oval Office. But interviews with a range of Democrats indicate that, if Mrs. Clinton is elected, she will lead a party that is unified around social tolerance, but sharply divided over how to achieve economic justice in an era of upheaval.