The best cure is to avoid the problem altogether. I usually build New Orleans and Chicago and another city just north of the Canadian border, give them plenty of culture (a great artist in Chicago will do the trick), and when the Americans appear they're usually so in awe of me (and the French unique power of entente) that they spontaneously vassalize to me. I once had the Americans found Toronto as their capital because when my Chicagoan culture rebounded in 1736, the Americans got sent to the closest uninhabited land north.
Short of that, if you want to keep New York or Boston or your Florida city, you should declare war on the Americans even before they found Washington. (it is possible to crush them with about 8-10 well-promoted infantry and some cannons sacrificed) Evacuate your troops in the flip area after the 2nd move so that they won't desert. Allow the flip and then use your cavalry to recapture the cities on the 3rd turn, and these troops won't flip.