You could try Publish settings > Flash > Audio Stream, and then changing that option to ADPCM. I've found that usually works to sync the sound in an .swf without any problems. The sound quality resulting from doing this varies somewhat, so it might not sound perfect, depending.
As for exporting the file as a video file like .avi, you might have to try adjusting the sound and video speed to sync things up by using another program entirely, like Adobe Premiere, Sony Vegas, or Camtasia Studio.
Alpha-Nuva
Go to File > Publish settings > Flash > Audio Stream (because I'm assuming you're streaming), and then set the bitrate higher (at least to a number equivalent to the fps you're using).
If you've already done this, then you may be in the same boat I was earlier this year. I had an animation that would go further out of sync as the movie went on. To remedy this, I simply staggered the frames (i.e. the sound came in a 1/4 of a second late, so I moved the animation down about 6 frames). It'll require a lot of guesswork, but it's the only other option.
Good luck and PM me if this helped.
vezanmatics (Updated )
I guess I'll try staggering the frames after the animation itself is done, though that sounds like a real freaking pain. Ah well, it's probably only a matter of pressing f5 a couple times. If it works I'll let you know.
Alternatively, it occurred to me that I could split the original flash file into its respective scenes, reducing the file size. Then I could render each scene individually and work some Aftereffects magic.