If you recently moved to Ohio, then you can legally carry with an out-of-state license in Ohio for 6 months after the move. See Title 1 Sec. 109.69.(B)(2). Otherwise, if you are a resident of Ohio for more than 6 months, then you can only carry legally with a permit from Ohio.
PS: I am not an instructor or lawyer. I am just an above average person who can find and read published law.