A drainage district operating under s. 88.16, 1989 stats., becomes a drainage district under this chapter as a matter of law on June 1, 1993. The records, assessments, funds and indebtedness of such a drainage district become the records, assessments, funds and indebtedness of the drainage district that takes its place. Before June 1, 1993, the circuit court of each county having a drainage district that has elected to operate under s. 88.16, 1989 stats., shall appoint a county drainage board under s. 88.17, if none exists, to take the place of the board of drainage commissioners of a district that elected to operate under s. 88.16, 1989 stats. The terms of office of the members of the new board commence on June 1, 1993.