All appointive officers in the service of the county and all elected county officers shall be deemed to be county officers for the purposes of this section. All county offices of an organized county shall be abolished when such county becomes an unorganized county. Payments equivalent to the salaries and other compensations customarily paid to the holders of such offices shall be paid to them until such time as their successors, except for the abolition of the offices, would have qualified and succeeded to their duties. Anyone re-elected to an office shall be deemed a successor to such office. If an officeholder possesses an indefinite term of office, payments shall not continue for more than one month after the county becomes an unorganized county. In the event that a person is elected to a county office which is abolished before the commencement of the term of office for which the person was elected, the person shall receive a payment equivalent in amount to one month’s salary of the office to which the person was elected.