On the mobile app as soon a message is opened it goes to Old mail which is very annoying.
Eliminate the OLD mail folder or keep opened emails in the INBOX
To have your inbox unified, please go to mail.aol.com through a laptop/desktop computer, click Options>Mail Settings>General then at Inbox Style select Use Unified Inbox style’’. Click Save settings.
So these settings reflects in the app, please uninstall/reinstall the AOL app.