It could be that your original video not just shakeable, but also has motion blur. For example if you shot with not enough light that the movement of camera make this motion blur. You can see this motion blur if you will see your source footage frame by frame (Alt+Arrows).
When you stabilized the video the camera motion is stabilized but the blur is on place. It looks like the video become unfocused on some positions.
You can’t do anything with this (at least I don’t know what can be done). I tried different stabilizers and they all do the same.
Also, during stabilization the video is cropped and upscaled a bit. This add some blur to every frame. You can try to deal with this blur by adding sharpness a little. Some stabilizers (DeShaker for Virtual Dub for example) can rebuild borders using information from neighbour frames. Unfortunately, Sony Stabilizer can’t do that.