For future searchers having this issue:
We are migrating these websites from another server with lscache. Because of that, the lscache directory in the user's homes did not have the correct permissions. We deleted all of the lscache directories and restarted lsws and this allowed the caching to start to work properly. However, for some reason, many of the sites' .htaccess files were not updated with the CacheLookup on line. Mass disabling LS Cache and then mass ENABLING LS cache caused this line to be added to the Wordpress installations' .htaccess files successfully. We're all good now.