How do I get into my Apple ID using my old Apple ID from my old iPhone

If I understand your question, you can't use one Apple ID to access other Apple ID.

If the email address for the other Apple ID doesn't work, have you tried to do a name change on that Apple ID?

Note the restrictions in the article.

Apple ID Name Change.        

 What to do after you change your Apple ID or password - Apple Support    I suggest doing this before you change your Apple ID or password.

I suggest you use the steps in the article below to remove your new ID from the phone. That will completely reset the phone and you can then use the desired ID to set up the phone.

Selling/Giving Away Device.      

Set up your iPhone, iPad, or iPod touch