At first I thought "I've seen this", because I had a similar out of sync audio issue on my Intel NUC5i3RYH which I had to resolve by installing dkms so I could update to the daily nose-bleed version of the sound driver library, but on further reflection, I think you are describing something else, because my NUC out of sync audio problem would effect ALL multimedia types (even YouTube videos played back in a browser window), but ONLY after the system was suspended, and then awakened from sleep.
If that doesn't sound like your issue because only a few specific AVI files are doing this, and no other media types are having issues, and suspend doesn't trigger the problem, then I still have an idea about something else that may be the root cause.
VLC 2.2.2 in 16.04 does have known sync issues playing some WMV files due to a stupid regression bug that effects how wmv files playback, so could it be possible that the files you are trying to play are actually WMV media mislabeled with AVI extensions?
You can check this using various tools in linux such as:
ffprobe -i name_of_my_media_file.avi
if the response includes something like:
Input #0, wmv, ....
... then you are dealing with a relabeled wmv file, which wouldn't be surprising because early on, thanks to Microsoft's total cluelessness about security, there were literally tons of security exploits possible with wmv files (like using them to launch malware web links), and when folks figured this out and started avoiding WMV and ASF file types, the bad guys figured out that they could simply relabel the files as something else like AVI.
If the files are not relabeled WMV files and are really AVI media, then it's possible that you have found yet another stupid regression in the libavcodec library which only effects certain AVI files.
If this is the case and you are dealing with legitimate AVI files that are simply getting hit with an issue due to a bug, then there are a couple things you can try.
First, would be to losslessly re-mux your files with FFMPEG:
ffmpeg -i 'my_media_file.avi' -acodec copy -vcodec copy 'my_fixed_media_file.avi'
If this works, you will end up with a new fixed AVI file that is the exact same size and quality as the original, but with the sync fixed.
May not work though, since FFMPEG and the libavcodec library that VLC uses share a lot of code, and this could potentially cause you to run into a similar regression bug issue.
So if the file is still out of sync, the only thing I could suggest, as a last resort, you could try booting into a temporary Ubuntu MATE 18.04 live-session from the an installer USB or DVD and see if you can play the AVI's without sync issues using 18.04's newer version of VLC (which has been updated to version 3.0.3).
If the file plays ok, then at least you can take heart knowing that the issue will be resolved whenever you get around to upgrading to Ubuntu MATE 18.04.