How do I eliminate and old email address which appears when I attempt to add an app

You wanna go into Settings --> iTunes and App Store. Log out and use your new / primary / current Apple ID instead.

You can use two different ones. One primary for your phone and iCloud and one for the iTunes and App Store.

Any apps you purchased or downloaded with your old Apple ID are permanently tied to that ID. If you really I want to switch to a new Apple ID, you will need to repurchase those apps under the new ID. This would apply to music and videos as well.

There is rarely a valid reason to change to a new Apple ID. If your email address changed, you can change your existing ID to use the new email. Then you would still have all of your purchased apps and content.