WASHINGTON—Three more top Democratic National Committee officials plan to leave the organization, including the chief executive and the finance head, as fallout from an email-hacking scandal further shook the party’s leadership ranks.

CEO Amy Dacey, communications director Luis Miranda and Chief Financial Officer Brad Marshall will be leaving the DNC, the committee said Tuesday. Their departures follow on the heels of Chairwoman Debbie Wasserman Schultz, who resigned last month after hacked emails that appeared to show staffers...