I lived 6 mo in NJ and 6 mo in NY and worked in NY all 12 mo. How can I find out how much of my NY tax was from the income made when I lived in NJ (double-taxed income)?

You need paystubs for this. Occasionally, NY employers will calculate NJ wages for NJ residents and place that on the W2 (this is because NJ wages are usually different than NY wages). If your employer only presents NY in the state section of the W2, you'll need to determine this with paystubs. 

Is your paystub detailed? If so, you may have "federal taxable wages" Year-To-Date before moving. Just keep in mind that NJ wages include the following additions to federal wages-

• Cafeteria plan/ IRC 125/ Café 125... health insurance
• Flexible spending, incl dependent care
• HSA contributions
• Pretax Retirement plans other than 401(k)- like 403(b)