Yep...ain't it ironic. Anti-shake makes it shakier. I have it too. Only on shooting video. Fine and works well for shooting stills. Just turn it off for video (it has a separate setting for video) and you'll be fine...but maybe a bit shaky.
Have you tried reducing the resolution of your video recording from the 1080p to 720p? I have found that the videos seem to come out a bit smoother at the lower res.
Damn, that was awful! One of the first things I do on any camera with electronic anti-shake is to turn it off! Heh, yeah I am not a fan of electronic anti-shake at all. I've always found that even if it smoothed out the shake a little there was always an affect on resolution or that it created other artifacts. I have a Sanyo HD1000 1080i camcorder and love the thing to death, but it doesn't have optical stabilization so I just leave the anti-shake turned off and try to be as steady as I can. When that camera reaches the end of it's life I'm only going to go with optical stabilized cameras. Since I don't shoot anything critical with my phone I can luckily live with anti-shake off.