can you post some examples? I use a 40D and can refer you to all of the above replies, if you are trying to focus on the eye and then recompossing to get the bird centre shot it is one shot focus mode using spot focus, AF is the way to go if you don't have time to mess about with focusing, if you do have time you are lucky
the other option is using a smaller aperture to get more DOF which increases your chance of getting the chest and eye in focus of course then you are really relying on good light to keep your shutter speed up. tbh from your thread title i would hazard a guess you are using AI servo which will be re-adjusting your focus constantly.
I am no expert of course,
