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Howdy,
Unfortunately, there isn't a simple way to upgrade CentOS from one distribution version to the next.
CentOS 5 is indeed old, but to get your sites running on CentOS 6 (or another distribution), you would need to setup a second server, and then migrate your Virtual Servers over to it.
There are instructions on doing that here:
https://virtualmin.com/documentation/system/migrate
You can't. You will need to re-install CentOS 6 from scratch then move your data over it.