There really are two basic problems for something like this:
1) Get all the data migrated.
2) Change DNS so it reflects the new server.
I would recommend that you solve them in reverse order in this case.
First go ahead and link your two servers with WHM's DNS clustering support. And then use WHM to transfer the data. This way traffic will go to the correct IP address from the start.
Once you have got DNS clustering setup, WHM has a built in multi-account transfer tool. That will let you copy all of the account data over from your old server. You can find some documents for this here: