“I left and others left because we were alarmed that Jeff would mismanage this organization as he mismanaged the campaign,” she said, expressing concern that Mr. Weaver would “betray its core purpose by accepting money from billionaires and not remaining grass-roots-funded and plowing that billionaire cash into TV instead of investing it in building a genuine movement.”

Kenneth Pennington, who was the digital director of Our Revolution, declined to go into detail about his reasons for leaving but confirmed that he was no longer with the organization.

The staff members who quit also said that they feared that the 501(c)(4) designation meant that the group would not be able to work directly with Mr. Sanders or the people that he had encouraged to run for office because such organizations are not allowed to coordinate directly with candidates.

Mr. Weaver did not respond to requests for comment.

In an email sent to Sanders supporters on Tuesday night encouraging them to participate, Mr. Weaver said that the new group would work together to “empower a wave of progressive candidates this November and win the major upcoming fights for the values we share.”

Mr. Sanders has been using his vast list of supporters to raise money for local lawmakers like Chris Pearson, a state representative in Vermont. He is also supporting Tim Canova, a liberal Democrat who is trying to unseat Representative Debbie Wasserman Schultz of Florida, whom Mr. Sanders accused of favoring Hillary Clinton’s campaign during the primary race.

In his live-stream announcement, Mr. Sanders sounded nostalgic about his presidential campaign as he repeatedly said his candidacy had changed the Democratic Party and pulled Hillary Clinton toward his progressive ideas.