Alien Registration number

Sorry - yes you do because it is on your Natz.

I assume "carry" doesn't refer to your wallet, but rather your identification.

You do, however, don't have to put that on any forms etc, unless it is explicitly mentioned in the context of being a naturalized citizen.
 
Your A# stays with you as a naturalized citizen (on your naturalization certificate) eventhough you hand in your GC at oath.
 
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