GH3 can also take your Nikon lenses.
I wouldn't buy a Blackmagic, since the rolling shutter is pretty bad, and inappropriate for documentary type shooting.
Thanks! I used the camera's optical stabilization (Steadyshot Standard, not Active). Then I used FCP X 10.1 stabilization in post for a few shots. I dialed back the power of the stabilization to about 50% of the default most of the time.