There are hundreds of men down there who are in the same predicament as you. I wont judge your actions like some, since I am not in your shoes. You will have to deal with the after-effects of whatever you decide.
But, short of killing yourself, leaving the country is the only way you are going to avoid going to jail.
Don't they eventually take away the right to get a passport, or revoke it? (I know this is a federal function, but at least in my state you apply for the passport at the very same District Court that issues the divorce decrees!).
If this gentleman is telling us the whole story I do feel for him. It doesn't sound like he is trying to evade reasonable support obligations.
The south must be really tyranical on this sort of thing. Where I live up north, I do believe it is a rarity for anyone to pay alimony beyond the first payment or so. The state does look with contempt on those who evade child support obligations, but I don't know that I've ever heard of someone actually going to jail. Losing driving licenses and prefessional licenses is common for hardcore deadbeats, but the orders are modified due to changing circumstances all the time.
Incidentally, I am helping a relative with her pro se divorce papers. She has children, and the father has not paid prior obligations (too jerky to hold a job), but we aren't going to try and drive him to suicide; that wouldn't be right.