No giving up the US passport. I carry both mine (British and US) with me when I travel. There are some countries that do have you claim one country or another, but neither US nor UK make you do that. If you applied for British citizenship, you are asked to take an oath to the Queen...but they don't say anything about you waving bye-bye to Bush.
When you enter one country with one passport, you are protected by that "citizenship claim". This is why, when you enter the US on your American passport, you are protected by the laws of the US, and given the benefits of a US citizen. If you use your British passport to enter, you are treated as a British citizen in the US. If you got into trouble, for instance, you would have to sort yourself out through the British Embassy. Same goes for the flipside. This is why you want to enter the US with an American passport, and enter the UK with a British passport (if you are dual).
BTW, don't get freaked out when your application asks you to send in your American passport. This is just part of the process; they do not confiscate it and you do get it back.
I can't for the life of me see anything but benefit having come from becoming a British citizen. I mean, I still had to pay taxes regardless. But now I can vote. I can buy property in the European Union. I can wave my British passport through the FastTracks.
Incidentally, you don't have to inform the US that you are dual. It's not really a change of status, it's just a bonus.
(I'm slightly confused, so I hope I answered your questions in a way that apply to you. Do you now have your British citizenship and now want a British passport, or do you still have to get your citizenship, and then will probably get your passport? After you get your citizenship, getting the passport couldn't be easier...the forms are at the Post Office; just send in what they ask for and it comes back to you all done within a week.)
Now go out and get your British passport, you knucklehead.