What can I do to listen to my music when the box pops up asking for my old e-mail and password and it won't accept my new one?
Most intelligent folks would enter the password for the account that is being prompted. Have you actually tried entering the password for the old Apple ID?
Considering that content acquired from iTunes is permanently tied to the account that it was acquired with, if you created a new Apple ID, entering that new ID and password will NOT allow you to play content purchased under a different account.
If you simply changed the email associated with the Apple ID, go to Settings > iTunes & App Store > select the Apple ID and then Sign out. Sign in with the correct email and password.