OK, one other question. When you say the "old ID", does that mean you created a brand new Apple ID and that the old one is still signed into Messages? If so, then the fact that the old email address no longer exists is not related to the old Apple ID. Do you remember the password for that old ID? If so, type it in when asked so you can get Messages signed back on, then sign it out of Messages. Once you get it turned on, you should see either a Send & Receive section or a Receive At section. Either one will then take you to a screen where you will see that old ID, and you can then click on it, and then Sign Out.
If you did not create a new Apple ID, but rather changed the old email address to the new one by EDITing your Apple ID, then post back.
Cheers,
GB