I need to add an old appleID to my iPad

I recently replaced my iPad when its battery got swollen. After I restored everything from the cloud (I forgot the password of the backup on my Mac), a couple of programs seem to be expecting my old AppleID which I had before Apple changed the rules on me a second time.

One such program is Marvin. It no longer thinks I have the full version. I could buy it again for my iPad (even though my iPhone thinks I have the full version). It's only $4.99. I told Marvin to Restore previous purposes, and it told me that I must be signed in with the Apple ID that was used to purchase the app.

So I'm figuring I could log off of my current AppleID and try to log onto my old AppleID, hoping that I can remember my old password (not likely), then open Marvin, restore, then log back into my new AppleID. If this works, I will write my old password down someplace in case I need it again.

Or is there a way I can set my iPad to have both IDs?

I'll probably just buy it again, but I'm trying to learn how this all works.