my phone shows an old email address on apple ID how do I change that

Sign out of that ID and sign in with the new ID.

Are you seeing this for App Store/iTunes updates?  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.