Name format change in new passport after tourist visa issued on old passport

[zainabbai]

Hi, I have tourist visa issued on my old passport in which the name appears as follows,

OLD PASSPORT (Visa issued under this name)

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LAST NAME: <BLANK>

GIVEN NAME: ZAINAB BAI

I now have a new passport as my old passport expired and the name appears as follows,

NEW PASSPORT

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LAST NAME: BANSWADAWALA

GIVEN NAME: ZAINAB

Please advise if I need to do anything or I can travel without doing anything by carrying both my passports together (This is my first time traveling to US). I am an Indian citizen.

Thanks!