Look in your "Documents" folder for a folder called "EA Games".
- Inside it, there are folders for each of the clients.
- In each of those, there is a folder called User Data.
- In each of those, is a folder for each of the accounts you have ever logged into from the computer.
- In each of those, there are folders for each shard the account ever logged into from that computer.
- Lastly, in each of those folders is each character's setting files.
However, doing them piecemeal is rather tedious.
The easy way is to
- Install the clients you want on the new computer and log in once in each, to create the client folders.
- Copy the "User Data" folders from the old computer to the corresponding client folder on the new computer.
This should work for both CC and EC - just don't accidentally put the CC User Data folder in the EC folder, or vice versa.