Final decisions of the administrative hearing commission in cases arising pursuant to the provisions of section 621.050 shall be subject to review pursuant to a petition for review to be filed in the court of appeals in the district in which the hearing, or any part thereof, is held or, where constitutionally required or ordered by transfer, to the supreme court, and by delivery of copies of the petition to each party of record, within thirty days after the mailing or delivery of the final decision and notice thereof in such a case. Review under this section shall be exclusive, and decisions of the administrative hearing commission reviewable pursuant to this section shall not be reviewable in any other proceeding, and no other official or court shall have power to review any such decision by an action in the nature of mandamus or otherwise except pursuant to the provisions of this section. The party seeking review shall be responsible for the filing of the transcript and record of all proceedings before the administrative hearing commission in the case with the appropriate court of appeals.