OK, so that is why I asked what you were trying to do. Updating and re-downloading apps do require that that Apple ID that purchased the app be the Apple ID that continues to provide the password, since legally, that Apple ID owns the media.
When you change your email address, unless the change, in any way involves an @me, @me, or @icloud email address as the Apple ID address, then you should be changing the address on that account to then new address. You have a single account. The email address is simply a user ID.
Sounds like you may have been told to set up a separate ID long ago, which is really not the way you want to go, because, in reality, you are setting up a brand new, separate Apple account.
Best advice I can give on that is to keep the password on the old Apple account the same as the password on the current Apple account. That way, if you are asked for a password, it won't matter which account it is for.
Cheers,
GB