Q: My iPhone keeps showing me an old Apple ID whose pwd I no longer know.

Did you update the old account or did you create a new account ? All content from iTunes is tied to the account that purchased/downloaded it, so if you've created a new account all content that you downloaded with your old account will remain tied to that old account, and only that old account will be able to re-download its content or download updates to its app.

If you created a new account then have you tried getting the password reset for the old one : http://appleid.apple.com, then 'reset your password'

Changed my apple id and the iphone 4S keeps asking me for my pword yet shows the old apple id.   Running out of ideas.

Tap on the old id in Settings > iTunes & App Store and log out of the account, and then log back in with the updated version of your account.