I have a lot of experience too with Linux and many other Unixes, so I do know what I am doing. I have even attempted a few times to upgrade CentOS5.x to CentOS6.x and the success rate was quite low (1 out of 3). Luckily they were all VMs, I didn't want to attempt it for real with a live system. There are far too many places to fail, and if you have a remote system you are taking a huge chance of rendering your whole system unbootable. If you need CentOS6, you are much better off and safer getting a new system with that version of the OS and migrating the data from the CentOS5 system. It is actually a lot less work that way and when you have everything working correctly on the new system, just change the DNS entries at your registrar to point to the new system so you have virtually no down time.
As others have mentioned your DA license would then need to be changed to a RHEL6 version to work properly.