If the charges were truly dismissed after you appealed then you were not convicted. However, there are many types of deals that people call "dismissals" that are technically not dismissals. I would call up the Chesterfield county clerk for the Circuit Court and ask them what the official disposition of the case was (hopefully their records go back that far without a trip to the warehouses) . If it was officially dismissed or "Nolle Pros'ed" then you can say you were not convicted. Otherwise, is a gray area that will require knowing more facts.
When you appeal a case from GDC to Circuit Court the prior conviction is reversed and the case is reopened. So once you appeal a conviction from GDC to Circuit you are no longer convicted until the Circuit Court says otherwise.
Simple answer is a Dismissal is NOT a Conviction. However, jail time to dismiss is an extremely unusual dispostion. You should consult your attorney from the 1999 case.
You should speak to your attorney who previously represented you. However, he has probably torched his files from 11 years ago. It is unusual for many attorneys to retain files that long.
If that is the case, I would call the Clerk's office in Chesterfield County and attempt to get a copy of the final order in the matter. I agree with the previous attorney. I don't see how you would complete 1 weekend in jail in order to dismiss the charge unless it's somehow disguised as a bond revocation.
In the event the charge is dismissed in Circuit Court, I would weigh my options. You WERE convicted of a crime, but the conviction wasn't final (you appealed before the 10 days expired). The only final order, if what you say is true, should be one indicating a dismissal.