As I understand it, a 5th degree felony in Ohio would not strip you of your firearms rights from a federal perspective. Federal law doesn't use the term felony, but specifies crimes that are punishable by more than one year in prison. Since an Ohio F5 is punishable by *up to* a year, it doesn't trigger the federal prohibition which is why you've been able to purchase firearms legally.
It does, however, prohibit you from getting a CHL as you've found out. You need to get your record sealed - I don't know much about that process but it isn't all that expensive, and as I've read here is something you can do yourself pretty easily. I'll let others fill in those details for you.