Do we need to carry any additional documents during international travel after updating name in passport ?

I moved to US on L1 Visa few months back. My first name and last name were in Surname in India passport because of which my Visa had FNU as first name and entire name as last name. I updated my i-94 to have the same name as per visa. My attorney mentioned that updating my passport should help me in getting rid of FNU and I got my new passport which has correct split name.

New Passport :

First name :

Last name :

Old Passport :

First name :

Last name :

Visa:

First name : FNU

Last name :

i-94:

First name : FNU

Last name :

I am traveling to India.

Are there any additional documents that I need to carry ?After re-entering US, will my i-94 be updated to correct name as per the new passport ?