@drgnslayr Looks like you cannot. I have wondered the same about my provider. So I googled "can I keep my Cox email" and this came up from the Cox support page on cancelling accounts:
Cox Email
Per our email policies, if you wish to keep messages and contacts after disconnecting Cox High Speed Internet service, you must move your Cox Email content to another email provider. After the disconnect is complete, we send you notification emails to your preferred contact email address on file to remind you to take action to save or move your Cox Email content.
We know switching email providers is never easy. Below are a few resources to help you move content from your Cox Email to an external account.
Note: To use the following tips, be sure to first set up your Cox Email in an email client program such as Outlook. Refer to Setting Up an Email Account in Windows for details.
In your email client program, export the files you want to keep to a local file. For example, Outlook uses the file format .pst.
Set up your new email account from your new provider in your email client program. Contact your external email account provider for assistance.
To gain access to your saved Cox Email content, import the local file to your new account.
Mayjay again: A couple of sites that Google brought up said that Cox used to let you keep your email for $10 a month, but those were a few years old and newer posts say they terminated that offer. In addition, some posts suggest that there might be a 90 day temporary period to keep it active while you transition to a new provider. But you should check with Cox directly.