Qualifications for this response ... support several entities with CentOS/Moodle servers remotely.
Some additional shared information might be needed to help.
1. Are the servers remotely hosted (RackSpace/Other) or locally hosted?
2. Are the CentOS/Moodle servers in a virtualized environment ... ie, VMWare/other?
Comment as to why the questions above: CentOS 6.4 is .2 versions behind. PHP could be higher as well.
The Virtualized environment adds a layer above CentOS/Moodle which could be the culprit. What does the hardware look like especially memory allocated? (note: when one looks at "top" for example, one is seeing the memory allocated by the hosting environment ... ie, the VMWare config of the server instance.
Sounds like the DB server is on the same box as the Apache. That true?
What is version of MySQL and what does it's configuration look like ... bottle neck might be the DB server and config of that.
Do you have mysqltuner.pl installed on the troubled server? IF the issue is NOT VMWare (it could be) then one needs to check for 'bottlenecks' that cause the slowness/un-responsiveness. One of those bottle necks might be the DB config itself - many factors could be involved.
Is Apache configured to use mod or fastcgi or other?
Have you checked the server logs messages/secure/access_log/error_log for clues?
With this sort of issue, the bottle neck/issues of un-responsiveness could be Virtual OS, Config of Apache/PHP/MySQL either individually or in combination with one another. If DB server is dedicated and Apache/Moodle on another server, could be a networking issue (network comes before application in the ISO model). In other words, there is no one simple answer. :| Case in point: spent 6 months with an entity going over the config of everything on a Virtualized (VMWare) CentOS 6.x/Moodle 2.x/PHP 5.5/MySQL 5.5.25 box that every hour at 32 minutes to 35 mintues after the hour would be slow ... even un-responsive. Turned out to be a VMWare issue ... another guest OS on that physical server had been given unlimited usage of the physical resources and when it did whatever it was to do, ALL other guest OS's on that server, would be un-responsive. Guest OS's ... still up, still running, guest OS not reporting any issues internally - but access to it, choked by the other system using up all resources.
'spirit of sharing', Ken