If any nonelective state office, except offices provided for in Article V of this Constitution, shall be vacated by death, resignation or otherwise, it shall be the duty of the Governor to fill that office by appointment. If the Legislature is in session, such appointment shall be subject to the approval of a majority of the members of the Legislature. If the Legislature is not in session, the Governor shall make a temporary appointment until the next session of the Legislature, at which time a majority of the members of the Legislature shall have the right to approve or disapprove the appointment. All appointees shall hold their office until their successors shall be appointed and qualified. No person after being rejected by the Legislature shall be again nominated for the same office at the same session, unless at request of the Legislature, or be appointed to the same office during the recess or adjournment of the Legislature.