Well there's your problem then.
Since avoiding camera shake is what you need to do, then you should be using Shutter Priority (TV on a Canon). You choose the safe shutter speed for the focal length you're using and let the camera choose the aperture. With your Sigma zoomed to 200mm your
minimum safe shutter speed is 1/200, and that's assuming good hand holding technique - if you can go faster with the shutter speed you should.
I found this link which gives you some idea of the technique. I have found better ones in the past but damned if I can now!
Click.