This could have been a total fluke, but I walked into Barclays with my passport and opened an account without any proof of address. I did have a job (I was registered with a temp agency), and the account advisor seemed more concerned with that than with me having proof of address, because she said my "economic viability" would determine what type of account she could offer me. Granted, I have an EEA passport, so that may have had something to do with it.
I would try popping into some high street banks and see who will work with you before buying the 1st Contact package. I wonder how they would be able to do more for you than you could on your own. Also, please do not buy the National Insurance number service! All you have to do is call your local Job Centre Plus and make an appointment to get one. Completely free.