Living on a mighty river (the Missouri) I know of a "sawyer" being a kind of snag, a downed tree whose branches "saw" back and forth in the current presenting a particularly dangerous hazard for boatmen, especially if the branches partially blocked the main channel. The tune is old enough (with that name) to have been popular among river boatmen prior to the Lewis and Clark expedition. I have heard it was one of the tunes that Pierre Cruzotte (the one-eyed fiddle player who accidentally shot Merriwhether Lewis on the return trip) was asked to play during the rare rest periods the corps managed.

I have no knowledge of the origins of Over the Waterfall and I'm sure there is no connection between the two but I DID learn them at the same time while on a river trip, with the San Francisco Folk Music Club, down the Sacramento river where we were asked to play nothing but songs relating to water. It is amazing how many songs of that subject there are.

CB

PS might be a fun thread!