The airline and the airport do not detain people. The officers with arrest powers normally detain people. The airline must give the passenger manifest to the U.S. Department of Homeland Security for screening against no fly databases which is reserved for suspected terrorists and sympathizers of terrorists. Normally, DHS does not conduct a scan of each person for outstanding warrants unless you are clearing Customs and entering the United States. As such, you have a low probability of detention.
However, as a former officer, I can tell you that I use to run everybody for open warrants even when my colleagues didn't. It is indeed the luck of the draw. It is always better to clear up any issues prior to travel. A disruption of your travel plans will be both embarrassing and an added inconvenience to you during the festive season.
If Georgia is not extraditing, that means they have not lodged the warrant as a "fugitive warrant" with other states, in which case law enforcement authorities outside of Georgia have no authority to arrest you. You are well advised to straighten out the situation with Georgia, but if your Virginia PO knows about it and hasn't had you arrested, that's a pretty good indication that no one at the airport will give you any trouble during your travels. In fact, considering the circumstances you stand much less chance of being detained if you fly then if you drive.