Don't know if you've had any help with this yet, but it's all to do with the optimisation in AS. In order to make it a smaller file, AS will automatically optimise the animation. I personally do the optimisation before I save it, so that I can preview the optimised version, and undo it if I don't like the way it's done it.
Go to file and Optimisation Wizard. Click next. In the next dialogue window, make sure the scroll bar is right at the top - at 'Better Image Quality'. Click customise - I use 255 colours, optimized Median cut, Error diffusion in the first tab, and in the Optimizations tab I have write minimal frames and collapse identical frames ticked. Press Ok. Press next - this will optimise the animation. Hit next again and a small preview is displayed - I find that's too small to see if there are any changes. So press next again, and it will apply it to your animation, where you can then preview it to check for pixilliation. If you don't like it, press undo. You can change optimized Median cut to optimized octree in customise if you don't like your result.
Once you are happy with the optimisation, save it as normal (file, save or save as). It won't optimise it a second time, because you've already done it.
Each animation will be different - sometimes median cut is good and other times octree is better. And sometimes, you won't get a result that you are 100% happy with. It all depends on the amount of animation and the positioning of the animation. There probably are better animation programs out there, but I've only ever used Animation Shop.