There's a few things to note here.
If you go to open a new account in a new bank with an overdraft facility, even if it is a student account, they will run a credit check.
There is no universal credit rating, so a 381 from Experian doesn't mean much. Each time a bank runs a credit check they will run their own scoring algorithms and make decisions in their own ways. This is important for a student account because a lot of new students won't have high credit ratings because they won't have borrowed any money before, short of a mobile phone contract, and banks are unlikely to turn student overdrafts down on this basis alone, particularly as they often ask for Student Finance dates, proof of university acceptance, etc. However, if you have defaulted on credit before, this will be taken into consideration as normal and you stand a higher chance of being turned down.
Each time a credit check is run by a bank for an overdraft, it will be recorded on your credit report. Apply for too much credit or apply too frequently and your chance of being turned away will increase as you'll come across desperate and lenders will realise that you've been turned away by other lenders.
If you apply for an overdraft with your existing bank, they may run an internal check rather than a full credit check, so this is worth a go before applying elsewhere really.
Finally, these things are best done in-person. If you apply online and get rejected, you have to start sending letters or calling up to try and explain. If you sit down with someone in a branch you can explain your situation, reasons behind any past credit issues, and ask for certain things to be taken into consideration. You can also gauge loan likelihood before opening an account to save bloating your credit report with dormant/very short-term accounts and credit check logs.
My advice would be visit your current bank in-person and speak to an advisor. If you're not successful there, speak to other banks in-person. Also bear in mind what financial support your university offers, and the Access to Learning Fund.