I’m not sure but looking at 3513.052 (G) it might be possible to withdraw your candidacy before the cutoff date and then seek another party’s nomination.
I haven’t looked too closely at Ohio’s laws, so I could be all wrong.
From Wikipedia: In United States politics, a sore-loser law is a law which states that the loser in a primary election can’t then run as an independent in the general election.[1] In most states, these laws do not apply to presidential candidates.