The first thing I would recommend is to get the latest version of codecs that your AVI files use. To find out what codec's your AVI files use, open an AVI file into Windows Media Player, click 'File' and select 'Properties'. Both the Audio and Video codecs are shown.
My recommendation would be to try a few different DVD authoring packages that can take AVI as the source. I have had this problem myself when trying to convert home-made AVI files (captured using a TV tuner) to MPEG. A few other examples of DVD authoring packages include DVDSanta (as suggested by saamil and probably the cheapest), Nero Digital, TMPGEnc, Ulead Movie factory, etc. If the audio sync issue happens throughout the entire recording, then it would be worth just encoding a minute's worth in each package to see how it turns out and then only try the full recording once you're happy with the test encoding. Use a rewritable DVD as the destination, until you get a successful encoding.
If the audio & video end up out of sync as you try other packages, then it could be something to do with the DirectShow codec, especially since a few codecs will not perform properly if not run at realtime speed. In this case assuming the audio & video are out of sync by the same amount throughout the entire encoding, you may be able to use TMPGEnc to resync the audio / video. Again, you'll likely need to perform a few test encodings to get it right.
