You do not need to enter the distribution adjustment on the New York return.
Remember that the credit is based on Tuition, Book and fees.
The distribution can be used to cover expenses such as room and board and transportation expenses.
When you make a distribution and part of that is taxable because the student received a scholarship or you receive an education credit, the distribution is taxable to the student, but there is no penalty tax.
When you decide how much to withdraw this year, remember that if there is a balance when the student is no longer attending school, the account must be rolled over to a family member or will be subject to the 10% penalty. For this reason, you might want to take the full distribution and have the student pay the income tax rather than leave it in the account and pay income tax as well as a penalty tax when the student is no longer in need. It depends on the account balance, how much will be needed in the future and if you would want to roll over the account in the future.
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