You actually must file for modification in the county where she currently lives. However, you need to retain an attorney if you want this to come out in your favor.
I am exclusively a family law attorney, practicing primarily in the metro Atlanta, Georgia trial courts. However, I handle appeals from anywhere in Georgia.
(1) Modifications have a snowball's chance in Hades without legal counsel.
(2) If she lives in Fulton, a change of custody must be filed there unless she agrees in advance to waive venue.
(3) Can papers be served at a courthouse? Yes, and they can even be served in a different county. Generally you will want a special process server to coordinate this
(4) You owe child support up to the date of a change of custody to you (if that happens), so keep paying, and if you owe back amounts, that will probably hurt your modification request. A skilled attorney may know how to weave the two cases together for negotiation.
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You have a lot going on and you need to hire an attorney. Your modification action needs to be filed in the county in which she resides. As to service. Based on your questions, service at the courthouse is bad service, however this is only if she objects to the service. As I stated, there are a lot of other issues, and this matter is more complex than you think. You need to hire an attorney.
Like the other attorneys who have answered your questions, I suggest you obtain counsel to assist you with your complicated matter. I would not suggest service at the court house the day of court. Typically courts like to see parties be provided time to respond and the court house is not a good place of service. You should file the modification in the county where she resides, but since you are unclear on her residence there are other options that an attorney could help you sort through. Best of luck to you!