Most banks that I know of use a system called Chexsytems where they report any unpaid accounts too.
So if you have a closed bank account with one bank and owe that bank money then you'll likely have a report on file with chexsystems that stays in their database for 5 years.
So unless you get lucky and find a bank that doesn't use chexsystems then you likely cannot open a bank account with a different bank until the 5 years have passed and you get removed from chexsystems.
However you might try an online checking account to get a checking account.
I'm not entirely sure but I don't think the online checking accounts check with chexsystems so you could try getting an online checking account instead.