Update: Some statistics and advancements for players were lost and were rolled back manually. PlayerData and chunk regions seemed to have avoided corruption, likely due to the server being smarter (not just blindly overwriting) with these files.During the aforementioned period, the server's disk was full. I was in the middle of changing the backup strategy, and while I was transitioning to the new one, I still had the old one going. So when it attempted to make a backup this time around, it filled the disk, leaving no free space for any new saves. I am currently unaware of any tools that can scan the world and find corrupted regions that have been updated to the 1.13 format. If you know of any, please let me know here in the comments or on the server.
A full, relatively recent and working backup of the server exists to restore any corrupted areas.