I am not accostummed to using the term "Motion for Commitment" and agree that it has more of a mental health connotation. I would suggest you use the term "Motion for Writ of Bodily Attachment/Motion for Commitment". I use the Motion for Commitment language in there to be consistent with the language contained in the Order, but what you are seeking is an arrest warrant. I am guessing that the Affidavit is what you would file to assert, under oath, that your ex did not meet the purge condition of the Order holding him to be in contempt.
As to your last question: that is a business decision you have to make to avoid all the travel expense you incur, and the knowledge that if he is in GA that law enforcement in FL will not locate him.
Without knowing the particulars, chances are that you will not get the relief you are seeking.
Still, consider this: if NC does not have jurisdiction over your ex, such that your domesticating the Florida decree in NC would not bestow upon the NC courts personal jurisdiction, then you are either going to have to travel for your future court dealings, either to FL or to GA. So, it may be better for you that he be disadvantaged equally as you by having to contest in FL, instead of only you being disadvantaged by having to travel to his new location.
Also, if he works for a national company that does business in GA and in FL, you can continue to use the Court system in FL for an income deduction order.
And, the cost to continue with the litigation in FL is much lower, at $50 to reopen the case, rather than starting up a brand new case in GA.
One benefit to the arrest warrant being issued in FL is that if he is stopped in GA by law enforcement, the FL arrest order should result in GA law enforcement holding him on the outstanding warrant.
But, none of that is going to get you support, if that is indeed the issue. For that, you may wish to consider using the NC court, which can then go through the process to have the court in GA start up enforcement proceedings, but for that you would need to utilize the NC state child support office.
Good luck to you.
I hope you found this response to be helpful. If so, your clicking "helpful" and/or "best answer" helps my Avvo rating and would be appreciated. This answer shall not be considered rendering legal advise but instead a general response to a general question. Avvo is a wonderful resource but nothing can substitute for an in-depth consultation face-to-face with a lawyer. The response shall not be deemed to create an attorney-client relationship, nor shall it create an obligation on the part of the attorney to respond to further inquiry from the questioner.