Charges have already been filed against you. At that point, the statute of limitations stopped running so that will not help you. I would imagine that GA authorities issued an arrest warrant. The chances are that as long as you stay out of Georgia, you will be safe. It seems unlikely that Georgia would try and enforce a warrant in Illinois for such a minor offense.
Most likely there is an bench warrant for you for failing to appear in Court for the underlying charge of Pedestrian Under the Influence. This means that if you did not appear in Court the judge issued a warrant for your arrest for not showing up in Court. Generally speaking there is a two year statute of limitations for this offense however charges have already been filed against you so the Statute of Limitations argument would not be a viable defense. While it may not seem like a big deal, it could come back to haunt you if you are stopped for a traffic violation or if there is a background check on this incident appears without a disposition and a failure to appear shows up as well. While I do not think Georgia would extradite you for this offense, if you are stopped for a traffic offense, the officer could/might very well take you into custody until this issue is resolved and the arresting agency is given the word that you are not to be extradited. This should answer your questions posted however If you have others feel free to contact me. Good Luck!
In order to keep a pending charge off of your record, it would be wise for you to engage a Georgia attorney to try and clear this matter up. Bench warrants can bite you in the butt at the worst possible time. While it may never come back to haunt you, if you ever do get caught, the time in jail won't be worth it.