Can Move my Outlook for MAC Cox email folders to Gmail?

I have been a longtime Cox email customer but am hearing so much about Cox no longer supporting email....on top of that I have a MAC and use Outlook for MAC as my email client.  

I have about 20 email folders in which I have stored a lot of personal stuff - and do not want to loose it!

Is it possible to move all of my Cox email folders to my Gmail account? 

A similar challenge has been that I cannot see my CURRENT Cox email folders on my phone or iPad when I access my Cox email....I only see a very OLD version of my folders from a number of years ago.  I have not been able to find Cox support who can help me sync my current Cox email folders to all of my devices.

Can anyone help me with this?

Thanks!!