Why do you need to apply for a new driving licence? You need not change it once you naturalise - it has nothing to do with your immigration status. Check the passport agency website for the one you will visit for the required documents. You can only make an appointment two weeks in advance, I think, so as soon as your oath date, or the day after, is inside that window, make an appointment. If none are available, you can also just show up. They will likely let you in with proof of travel.
I should also mention that you must be let in without a US passport if you are a US citizen, but it will be a huge hassle, and they could fine you. I would avoid it, unless I were white.