Have your mail forwarded to your new address at the Post Office and change your address on your driver's license. Pretty sure you can do it online and save yourself a trip to the DMV.
A lease or some type of mail with your name and address on it... You also have 45 days to change your address but no one really follows that... If you moved and all you want to do is change your address you can always go online and do a change of address if it's just a move.
Take your lease and some form of proof for a utility connection. The DMV will tell you what you need from there.
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