Logging out of apple id w/o knowing password

I have two apple id's. One of the id's is associated with an old, no longer active, email account. This is the account that i am signed into on my iPhone. However I want to use my most current apple id (i.e. one that has my current email address associated with it). The dilemma I face is that when i try to sign out of my apple id on my phone that has the old email address; it asks for me to enter my password. I don't recall that password and do not have it written down anywhere. Then when I try to recall the password online, it asks for my birthday, but when i enter it i am notified what i entered is wrong (even though it is technically correct) - a myriad of issues, i know. Furthermore, i cannot have apple send an email to reset the password to that email account because it was disabled because it was compromised.

With that said, how can log out of this id with out being able to reset, or remember the associated password? Or is there a work around/solution I am overlooking