Management Site Shows Content Changes but Delivery Site Won't Until Memcached is Flushed on All Servers

Recent Changes:
We recently installed Memcache on the servers and the MemCache API module on Drupal.

Background:
To give some background with the current site configuration, we have a single server running our site "management" where all the site content is updated and then we RSync Drupal and all, to an NFS Mount where it sits on "delivery" servers to load balance traffic to site etc. There are other layers like a loadbalancer and varnish servers sitting in between the loadbalancer and the drupal delivery servers.

Request for Support:
We don't believe our configuration may be entirely correct as memcache functions as expected, storing pages, we can view memcache statistics (hits misses etc), but when the content managers update the site, it is only reflected on that management server and won't display on our delivery servers until the memcache service on all of the servers is restarted at the same time. ONLY FOR UPDATED CONTENT. New pages created show up as expected on the delivery servers.

We think we have missed some sort of configuration for Memcache or perhaps in our settings.php to clear the cache of pages after they are updated so that the delivery servers aren't still delivering old content.

We have confirmed it is not a varnish cache issue because the issue persists if hitting the servers directly, bypassing varnish.

Drupal Core Version is 7.97
memcached Version is 1.6.23
Memcache API Module is Version 7.x-1.8
PHP Version is 8.1.27
Drush Version is 8.3.3

Copy of settings.php is attached with the names of some values changed but configuration is the same.
Copy of /etc/sysconfig/memcached is also attached.
Image of Status Report is Attached