If you are getting married in London in the next month or so, you need to start taking care of things now. You can get married in any that you want, pick which registry office you want to get married in. Call them and make a booking for your ceremony.
Then, call the council in which you and your DF live, and make a booking to give notice of your intent to marry. You have to do this since you are subject to immigration control. Do this
NOW. I called my council two months in advance of my wedding, and the first available appointment was a week AFTER my wedding. Needless to say, I freaked out and ended up begging a lady in the Registry Office for an earlier appointment. Luckily the woman in charge was a sweetheart and took sympathy on me and the fact that my family from the States had already purchased flights and all...
You are right that you have to have lived in the borough for more than 7 days before giving notice. You can prove this using council tax bills, utility bills, etc. The passport stamp is not sufficient proof because it doesn't state in which borough you live.
Once you give notice, 15 days must pass before your ceremony can take place. You have to go and pick up the certificates for marriage in person. You then take these with you on the day of your ceremony.
Please note that once you have given notice, you cannot change the venue, unless you go through the entire process again.
Hope that this helps. It took me a while to figure it out! Congrats on getting married!