I believe you are on the right track with the completion of FOA, but I think what you are saying is more geared towards misedemeanor crimes. Even completed FOA for FELONY charges will show up on a criminal justice run RAP SHEET. They never go away unless expunged.
Felony offenses in general are also eligible for first offender. The law does specify certain felony offenses that are not eligible, but does not disqualify all felonies.