What mail client are you currently using with the new cox account?

Generally speaking, you would configure your mail client eg. Live mail with both email accounts. Then you would export mail from the old account then import it into the new account.

Importing and exporting mail in Windows live mail.

You should be using a newer mail client such as Outlook or even Thunderbird mail, if you go with Thunderbird as your new mail client, I would first configure both email accounts then move forward from there.

You would be able to simply move mail between accounts once configured in Thunderbird.

Cox email settings for Thunderbird if needed.