Voice mail

I would like to use my phone base unit for voice mail instead of COX. It's easier to access and see if I have any voice messages. However when I turn off COX voice mail my base unit voice mail does not give the greeting or offer the caller a chance to leave a message.  Worked well until we were set up with the modem that included phone. Is there as way to do this? I have an AT&T DECT 6.0 phone with a Digital answering System.