You are double-taxed on the overlap. New York does not exempt this income from NY tax. What NY will do is issue you a tax credit for the amount of tax you pay to Connecticut for the income you earned in CT while still living in NY.
If TurboTax does not calculate the amount for you automatically (it may be difficult for the program to do so given the situation), you will need to determine the amount of income you earned in the overlap before moving to CT. Then, prorate the amount of CT tax to the percentage of the CT income that is the double-taxed income. To determine this, divide the CT income earned while still living in NY by the amount of overall income reported to CT. Then, multiply this percentage by the difference of the amount of CT tax withheld minus the CT refund or the CT additional tax plus the additional amount due. To illustrate:
Suppose the amount you earned in CT for the remaining portion of the year was 25,000, and the 2-month period was 5,000. The double-taxed income was 20% of your CT income (5,000/25,000). Now, let's say that on the 25,000, you had 1200 withheld and a 200 refund. You would multiply 20% (the double-tax percentage) by the net tax of 1000 (1200 minus the 200 refund). Your CT income would be 5,000, and the tax on the 5,000 is 200. In this hypothetical scenario, these would be the numbers you would enter on the New York part-year resident return for "Taxes Paid to Another State". But NY will not exempt that income from NY tax.
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