before you leave US, you can request a duplicate copy of your US license @ DMV so that when Canadian DMV will exchange your license, you still have one for you.
You could do that but unless you are from Florida, most U.S. states are not going to give you a license if you tell them you want a replacement because you surrendered your original one to a Canadian province. You can lie and say you lost it, but that is against the law. It's called misrepresentation. You might never get caught, but if you do it could put your immigration status into question I imagine.
In my situation, I spend most of the year in Canada and part of the year in my old U.S. state. Legally, I am probably a "resident" of both places, so it would be convenient to have a license from both places, but it doesn't seem possible w/o committing some form of misrepresentation. Unless someone can come up with some way this is possible?