Can a couple become legally married by living together as man and wife under the state’s laws (common law marriage)?

No. North Carolina does not allow common law marriages. In North Carolina, you must either be solemnized as husband and wife in front of an ordinated minister of any religious demonization, minister authorized by a church, magistrate, or in accordance with any solemnization recognized by any religious demonization, or federally or state recognized Indian Nation or Tribe.