Ammending the state returns and State wages inputs

My wife lives in New Jersey, but works in New York.   With the initial return submission via Turbo Tax, she allocated a large portion of her income from New York to New Jersey since she was working from home during the pandemic.  However, after submission and review, New York indicated that working from home does not justify allocating income to New Jersey.  Therefore, she will need to amend both the New Jersey and New York returns.  Can you please explain what process I need to follow to submit an amended return for both states?  Do I need to get prior authorization from the NJ and NY divisions of taxation before I submit?

 

Also, I am questioning my wife's state wages inputs.  In Turbo Tax, I keyed in the New York State workplace W2 info with the wages on line 16 and income tax on line 17.   However, the New Jersey residence W2 shows a wage amount on line 16 approx 5,000 more than New York.  I wanted to confirm if I just add the incremental 5,000 amount to the New Jersey line for #16 wages, so the total of New York plus New Jersey wages equals the amount of wages on the New Jersey Residence W2 form?

Thank You