How do I remove my old Apple ID -email ? It keeps popping up instead of my new.

If you have changed your ID or password, have you followed the steps in this article?

What to do after you change your Apple ID or password - Apple Support.      

Check the App Store and iTunes Store to see if there are any updates for apps/purchases. A request for another Apple ID password is often caused by updates to apps/purchases that were made when using the other ID. Those purchases are tied to the other ID and can't be transferred. One option is to delete the apps/purchases made by the other ID and download them using the new ID if desired. If necessary, check your purchases history to see which apps/purchases weren’t made by the current ID. 

Purchase History Review.   

Or you can try to set up Family Sharing  between the IDs to keep access to your purchases.

Family Sharing.         

Family Sharing - Set up.        

You are welcome.