I don't remember anything about carrier. Just username and password, and username is an email address. KB32514-How to change or update the BlackBerry ID Username on the BlackBerry smartphone
Do you mean you used the hosted Verizon BlackBerry email address as your username? If so, that wasn't a good idea, and you don't want to repeat the bad practice by using the hosted T-Mobile address. Use something like a Gmail or Yahoo or Outlook address, or your ISP email address.
If I'm not understanding, explain. My advice is edit the BlackBerry username with an appropriate email as outlined, and then you should be prompted to log in on the device. If you are not prompted right away, open BBM or BlackBerry World and that should launch the prompt to log in with Blackberry I'D. If not, we will troubleshoot that issue.