On September fifteenth in each year when a state election, as defined in § 9-1, is held or not later than September twenty-second in any year in which the date of a primary for nomination to a state or district office is advanced pursuant to subdivision (2) of subsection (a) of § 9-376, the Secretary of the State shall mail to each town clerk a list of the names of all duly nominated candidates for state and district offices to be filled at such election, filed in his office prior to such date, for whom an elector may vote, with the respective party designation of each of such candidates. Forthwith after the results of the primaries for state and district offices are tabulated, the Secretary of the State shall mail to each town clerk in the district the names of all duly nominated candidates for such offices, with the respective party designation of each of such candidates.