Old Apple ID keeps popping up asking for password - have changed email servers and old one no longer available. How do I get rid of old ID?

Hi. If it's really an email account, you can delete the account under Settings > Mail. If it's an Apple ID, maybe the one used to setup the device, you need to enter the password for that Apple ID. An Apple ID doesn't cease to exist, even if the email account used as its name goes away. Many people create new Apple IDs for new email accounts, instead of just changing the old ID to use the new email as its name, see Change your Apple ID - Apple Support.

You've probably got applications installed that were purchased/downloaded under the old Apple ID. You'll need to remove and redownload them.

You will need to log into https://appleid.apple.com/ and change your credentials..

it is still asking for 10 year old email password, and that account is not on my iTunes account or my device.

If you are being asked for an account's password it implies that you might have apps from that account on your device