Remove the NY income as duplicate wages. For your NJ part-year return, you only pay tax on the income you earned while living in New Jersey. You remove the NY wage amount so that NJ tax is calculated against NJ income.
Your double-taxed income is also the same $60,000. New York City should not have been taxing that portion of income, but only the portion on the $30,000 you earned while living in New York City (NYC only taxes income earned in NYC by residents of NYC). The amount of tax paid to New York City on your New Jersey income is $0. (Even if your employer mistakenly withheld NYC tax after your move, the mistaken tax will be "refunded", or at least calculated into the NY state return so that, in effect, you do get any mistaken amounts refunded. But the actual NYC tax on your NJ income is $0).
I really liked your example. You broke it down well.
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