Party affiliation can only be changed at a Partisan Primary Election.
O.R.C. 3513.05 ¶7
"For purposes of Ohio election law, an individual affiliates with a political party by voting in that party’s primary election." "An elector is considered to be a member of a political party if they voted in that party’s primary election within the preceding two calendar years, or if they did not vote in any other party’s primary election within the preceding two calendar years."
Per - Election Official Manual - Chapter 13: Party Affiliation - Directive 2016-26"