In the event of a vacancy at any time in any of the offices of any county of the State the Governor may appoint some suitable person, who shall be an elector of the county, and, upon duly qualifying according to law, he shall be entitled to enter upon and hold the office to which he has been appointed:

(1) If it be an elective office, until the next general election for such office if the term of such office be fixed by the State Constitution or until the next general election if the term be not so fixed, in which latter case an election shall then be held to fill the unexpired term and in either such event such person shall hold office until his successor shall qualify; and

(2) If it be an office which was filled originally by appointment, until the adjournment of the General Assembly at the regular session next after such appointment.

Any officer so appointed shall be subject to all the duties and liabilities incident to his office during the term of his service therein. Any officer elected to fill an unexpired term under the provisions of this section shall hold office for such term and until his successor shall qualify.