1. You need a lawyer. If the ATF says you can't pass the background check because of the drug charge, that means (in their opinion) it's illegal for you to even possess a firearm. Seriously, talk to a lawyer.
2. Assuming your facts are correct, the FBI messed up. An Ohio F5 is punishable by not more than one year. Therefore it is not a term "exceeding one year" and therefore not a federal disqualifier. Still, if you appealed and the denial was upheld, you need a lawyer.
That may not be the kind of help you were looking for, but there it is.