Overcast days are not good for shooting moving subjects at high shutter speed, so you may need to change your expectations. The suggestions above are all good and should improve your results, but I'm going to suggest something completely different and it doesn't require new equipment. Use Tv mode and set the shutter speed to 1/50s or slower and try to capture some motion blur. The objective is to track the subject during the exposure to get the head as sharp as possible but not to worry about anything else. Blurred wingtips or a streaky background give a sense that the bird is moving at speed. The technique requires some practice, and the results may not be to everbody's taste, but I quite like the effect it can produce.
An example:
Black-Browed Albatross taken at 1/30s, f/7.1, ISO 100 by
Andrew Stringer, on Flickr
This works best with large birds that don't flap their wings very fast.