Impeachment doesn't prevent you from running.
The disqualification language in the Constitution around impeachment is "disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States"; Ross Garber, an impeachment and political investigations lawyer noted that no court or Congress has ever settled the question of whether the presidency counts as an "Office of honor, Trust or Profit under the United States" from which the Senate is able to ban an impeached and convicted person.