Whenever all of the owners and other persons having a vested interest in those tidelands or shorelands embraced within any plat of tidelands or shorelands or within any portion of any plat in which there are unsold state-owned tidelands or shorelands, shall file a petition with the department accompanied by proof of service of the petition upon the city council, or other governing body, of the city or town in which the tidelands or shorelands described in the petition are situated, or upon the legislative body of the county in which the tidelands or shorelands outside of any incorporated city or town are situated, asking for a replat of the tidelands or shorelands, the department is authorized and empowered to replat the tidelands or shorelands described in the petition, and all unsold tidelands or shorelands situated within the replat shall be reappraised as provided for the original appraisal of tidelands or shorelands. However, any streets or alleys embraced within the plat or portion of plat, vacated by the replat shall vest in the owner or owners of the abutting lands.