Some Boissonneau-dit-Saintonge did drop the Boissonneau part and kept St-Onge, however this was done mostly in the County of Deux-Montagnes, north of Montréal in the 1700's. To this day, this branch of the Boissonneau family still carry the surname St-Onge. You might also have a few Boissonneau who switched their name to St-Onge in the States, in the late 1800's to simplify matters, but they are the exception.
The St-Onge in the Gaspé area are mostly Payan-dit-St-Onge. Some kept the namePayan and some the name St-Onge.
Saintonge per se was never a surname. It was given as a "dit" addition to people who emigrated from the province of Saintonge in France, simply added to their name. There were approximately 25 surnames in New France that had "dit St-Onge" added to them. The largest group would be Payan.
The former French province of Saintonge is now called the Department of Charente-Maritime.
You'll notice that I sometimes write Saintonge without the hyphen, the reason being that the former province of Saintonge was written that way. The word "Saintonge" came from Latin. In the Christian tradition, there never was a Saint named Onge. People started writing it "St-Onge" because it sounded like the name of a Saint, such as St-Pierre or St-Jean (surnames well known in Canada).