iPod version 9.3.5

I've changed BOTH my Apple ID and my password. My iPod is still showing the old ID, AND I do not remember the old password.

1. When I go to Settings on the iPod, the iCloud account still shows the old ID next to my name and the icon.
2. When I tap <name, ID> under iCloud, the next screen shows "Apple ID" and the old ID.
3. When I tap "Contact Information", the next screen shows the icon and my name at the top with the new ID.
4. I cannot sign in to iTunes from the iPod because it's linked to the old ID and old password.
5. I cannot "Sign Out" of the iPod as advised in previous threads, because it wants the old password.

I would very much appreciate advice! How can I get my current Apple ID and associated password onto this iPod?