That's an awful akin. It explains how to export previously-received email and contacts from their server...not your email client. Yes, you can keep recorded messages on your telephone...just not voice mail on their server. Yes, you can keep email in your local .pst...just not on their server.
If you can find Email-Only Service on Cox.com (you won't), order it.
"Upon disconnection of your Cox High Speed Internet Service, Cox will delete your email address, mailbox, all email messages, and the associated content from each Cox email address associated with your account within 90 days of disconnection."