Welcome to the forum and congratulations on your new A200!
I also have an A200 and I noticed the auto WB doesn't seem to be as "automatic" as it could be. I too noticed washed-out pics with the flash. I compensated for that by bumping down the flash exposure a few steps and that helped a lot.
I'd like to think that for the most part the auto WB is doing its job. But in other situations it fails to automatically adjust the white balance at all, or at least not enough to save the picture. For example, indoor shots with indirect incandescent lighting produces horribly yellow pictures unless I manually set WB to the Tungsten setting (where the auto WB should be doing that for me, or at least making a valiant effort to compensate).