The Verizon Email is also your login to pay your bill, and access your Calls and Voicemail Messages, and to access TV online services. So they tie so deeply to your account, that the only way to get rid of it is to cancel service.
Work arounds include:
- Ignoring it
- Forwarding the messages to your new email
I can't think of much more than that. So unfortunately you are stuck with it.