Since you have joint custody, your wife cannot change the residence of the children without the permission of you and the court. Since you have custody of one of the children already, she certainly cannot take that one out of state. In general, courts are disinclined to separate childre from their siblings. You need to get an attorney quickly so that the laws of your own state can be applied.
I agree that your wife can move but that does not necessarily mean she can move the children to AZ. The court makes the determination what is in the best interests of the children, moving to AZ with Mom or remaining in GA with Dad. The court's analysis is more in-depth than what I just wrote but there is no presumption that the custody remains with the custodial parent if moving out of state. There are several ways you can approach this matter and I recommend that you meet with an attorney and discuss the options of litigating a modification of custody; or mediating a modification of custody, etc. I hope this helps! Thank-you, Melanie Brubaker
This move is grounds to seek modification of the custody order. The court will then make a decision that is in the best interest of the children. A complicating factor is that the 17 year old has the right to make an election, and the court will almost certainly respect that choice.
You should consult with an attorney who can evaluate the specifics of your situation, and advise you accordingly.