However, the statute then prevents a person from filing for an office in the general election as a write-in candidate or with a "nominating petition". If Joy is appointed to the seat by party leaders, then this provision clearly does not apply. The only question is what exactly is a nominating petition.
Under well established Ohio law, a nominating petition is a petition filed by an independent candidate for office. If Joy would run in a special primary, she would instead file a declaration of candidacy.
This issue is ripe for litigation but, in my opinion, will likely break in Joy's favor.
It seems to me that being appointed as the nominee would fall outside the scope of the statute. I would also posit that "sore loser" statutes are meant to avoid, for example, Lieberman running as an independent after losing the Democrat primary (obviously CT doesn't have such a statute), and thus that Padgett would only be prevented of running as an independent for Lt. Gov. (although I must admit that I have not read the text of the OH statute). I recall that a similar issue came up in FL in 2004 when the Democrat nominee in the 22nd CD, Jim Stork, dropped out of his race against Clay Shaw for health reasons; some Democrats propose that outgoing Congressman Peter Deutsch, who had lost the gubernatorial primary, replace Stork on the ballot, but others suggested that the sore loser statute would prevent that, and in any event Deutsch wasn't interested.