Are you using an email client (like Outlook or Windows Mail) to access your verizon email? Or a browser based client? If the former, then it should be the case that all your email is stored locally on your computer as it gets downloaded from Verizon. So it's not going anywhere if you switch to Yahoo. If webbased, you're not going to be able to migrate it to yahoo without forwarding all of it, there's no way to "upload" your email to another service. You also may want to consider switching to Gmail (From Google) or Outlook.com (from Microsoft) rather than Yahoo.