Flying the nest for the first time and looking to set up a new bank account for bills

If you'll be paying utility and telecom bills by direct debit, a Santander 1-2-3 Lite will give you cashback on them. The Lite part is important, the full fat version has a £5/mo fee, whereas it's only £1 which you will easily get in cashback. For short term savings, TSB Classic Plus at 5% (on up to £2000) might be a good option for you.
One option would be to use TSB as your "main" account and keep your overdrawn NW account close to the limit, as long as the overdraft remains interest free. Cycle some money through NW after every payday to ensure you're meeting the conditions, but otherwise don't use it. Save up money and make sure you're ready to clear it before it starts costing you. On graduate accounts the bank will often phase out your free o/d by reducing the limit. You should be prepared for this.