I've been admiring both of these cameras, but here's what I can't figure out: The 370 is obviously designed more for shoulder mounting, but comes with a non-stabilized lens. The AF100 could (possibly) benefit from my OIS Canon zoom. So which combo is likely to produce steadier handheld footage?

Philip.
No real way to know without testing and different people could come up with different results on this one.
The 370 is ergonomically much better for handheld as shooting off your shoulder for extended periods of time is a relatively predictable look.
The AF100 is set up as a chest camera when handheld (unless you buy accessories) so you may find it harder to use over time that way.