I lived in New Jersey for 20 years, but relocated to New York on 9/1/2013. I still own my house in NJ, my car is registered there, and almost all of my business for 2013 (except health insurance) is based in NJ. For the last 4 months of 2013, I rented in NY. So, for year 2013 tax purposes, do I live in NJ or NY, or do I have to file both?
You're most likely required to file in both for 2013 - since you've lived in both. From 2014 and on you're definitely a NY resident (since you're renting a place there and live there), and you may very well continue being NJ resident (since you're essentially continue being domiciled there).
I suggest talking to a EA/CPA licensed in NY and NJ to try and see what you can do to avoid being resident in both the states, or see if it is at all an issue other than filing everything double.