I think the Sony stabilzer is based on Mercalli, which is what I use. The process of stabilizing requires zooming in if you don’t want any borders, i.e., if you want the video to continue to fill the screen. It has to move the video around to offset unintentional camera movement, and it zooms in to avoid borders.
I don’t know whether there is a border option available in the Sony stabilizer, but in Mercalli you can choose to have it make a border. In that case I think it doesn’t have to zoom (never tried it, so I don’t know for sure), but you end up with borders around the video — it doesn’t fill the screen.
As far as the graininess, it may just be a result of zooming in, thus reducing resolution. If find that if my 1920×1080 HD camcorder video is quite shaky, it starts looking more like DVD quality if the stabilizer had to zoom in a lot to steady it.