It would be helpful to know the exact camera settings you used while shooting. Were you using any auto functionality of the camera, or were you shooting fully manual? Were you shooting handheld, or with a tripod? If you were shooting handheld without the "SteadyShot" function enabled this could be the reason why your video looks jittery. The effect that you are seeing in the text of the video may also have been caused by the "SteadyShot" function if that was turned on.
It is very difficult to give you an accurate assessment of your footage without seeing any of it. Can you post some of the video online for review? Best way to assess the video is from the camera original footage using your camera as a playback device. If it looks strange in-camera than it is most likely the footage, or the camera itself. After that, move over to your computer and view the camera original footage with a self-contained playback application like VLC.
Two different SONY cameras had the same results. I do not like the jittery look of 1080p. Is it possible to convert 1080p to 1080i after the footage is shot?