HOLLYWOOD, Fla.—Greg Shesney had a campaign sign for Rep. Debbie Wasserman Schultz on his front lawn through most of July, broadcasting his preferred candidate in the coming House Democratic primary.

His sign now sits in his shed, taken down last week after Ms. Wasserman Schultz was forced to resign as Democratic National Committee chairwoman amid charges that the DNC sought to undercut the presidential primary campaign of Bernie Sanders against Hillary Clinton.

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