I think I would fly in to South America, then illegally hop a border in to Brazil and live as an illegal immigrant in a large city. Brazil is big enough to get lost in but has enough expats that you wouldn’t draw too much attention.
Well who is trying to find you? The NSA? Is your picture on CNN for molesting one million babies or something? Yea that is a hard one to crack and I’m not sure I have advice.
For less levels of government and attention shit move to another city, or follow even sven’s advice. Hell get married in Brazil and then get permanent residency(I have no clue about Brazil’s immigration law but going off my own experience the USA is not informed or involved) and then once you have that you can work legally.
A little bit of wheeling and dealing with the Devil can facilitate things.
See: Damn Yankees. (ETA: Or: Twilight Zone episode Of Late I Think of Cliffordville.)
Mistake number 1: announcing you’re trying to disappear in a national magazine. So don’t do that. 
Hell, I practically live like that now.
There was a movie about this a few years ago, called “Erasing David”. IIRC, it took about 2 weeks for him to be “located” after he “disappeared”. He just wanted to see if it could be done, and if so, for how long.
That was a fascinating article, though.