If you open an account at the same bank where the ODs are they may well take all of the money you deposit to cover the ODs.
If you open an account at another bank I'd suggest you take the bank with the ODs cash to pay the ODs.
You may have a problem opening an account with another bank though. Many banks report account status to a company that then report it to other banks. Sort of like a credit check.
If this happens and you can't open an account and you need a bank account for something like DirectDeposit you might want to look into something like a GreenDot or American Express Serve account. You can get them at WalMart and other retailers or directly from their internet sites.