This response is intended for informational purposes only and is not intended to establish an attorney-client relationship.
You are using the term "thrown out," as many people do, but that, of course, is not a legal term. What you need to do is go back and look at the record of that DUI case and see what the final disposition was and what the Court's order was. If it was dismissed or if the state chose to nol pros (nolo prosequi meaning the state chose not to prosecute you) great. Otherwise, if the matter was placed on the dead docket, for instance, it will still show up on a criminal background check. In fact, if you were arrested, but not convicted, for DUI, that also could show up on a criminal background check unless you applied for - and were granted - an expungement of your record. However, since your attorney in 2006 saw the charge, I doubt it was expunged.
Your first move is to obtain the documentary record of your DUI case from the clerk of the court where it was heard. This may take a little time since the ticket is from 2003. From there, you should go to an attorney and review the record, not only to see what happened in 2003, but also to see what, if any, remedies are available to you now.
An employer generally will get a copy of your Georgia criminal history. If you were arrested or cited the records will reflect that ( and the disposition) a nolo only helps against license suspension. The important thing to ask the employer to note is the final disposition. Essentially without an expungement your record will show all violations.
A couple things in response to your response. In addition to what I have already said, don't confuse a PLEA of nolo CONTENDERE (no contest) with the State's decision to nolo PROSEQUI (not procede with the charges) those are two different things that only someone w/ a Classics degree (like me) might immediately notice, but any lawyer shoudl also know the difference.
I understand the improper paperwork situaiton, I just am suspicious that the case was not resolved quite as favorably as you think since your 2006 attorney knew about it. That said, even most attorneys don't realize that, even when you have an arrest or conviction EXPUNGED from your record, my experience has been that the arrest is visible on a background check, but simply notes that it was expunged, such that potential employers, etc., will take that as an indication that it was baseless or erroneous and not take any such record into account.
I am not at all certain that driving charges fall off of your record automatically after a certain period of time a la past due payments on your credit report. Even speeding tickets are criminal chages and the law in the State of Georgia, for instance, is that the penalties for a second DUI conviction in ten years are far stiffer than a first, so I definitely would not count on this matter, however it was resolved, simply to fall off of your record.
It is best ot do as you said you would do and consult an attorney about your options. With regard to what "a while" is, that statement was vague on purpose ... I am not sure how long the Clerk of the Court you are dealing with will take to retrieve such records, but if I had to guess, I'd imagine it would take a week or two to get the stuff out of storage, etc., assuming the records are not stored electronically and can be pulled up immediately.