I did this when I moved. You can do the change of address on the DMV’s website, but you have attach a proof of address along with your online application (a utility bill or something). Like honey_badger, it took about a week to get my new lcense and a few more days until I got my new RPP sticker.
In the meantime, you can get a visitor’s parking pass from the police station if one of your housemates have some kind of proof of address for your new parking zone. They show their DC license or mail with your new address and they can get you a visitor parking pass for your new parking zone. The passes are good for 2 weeks but you can go back after 2 weeks and get another one if you need more time.