Yes, this is a problem because Turbo Tax does not have the ability to handle multiple move situations. Here is what I suggest.
Determine the amount of days lived in both states. For the amount of days you lived in NJ, you can say you lived there from Jan 1 up to the amount of days. So if you determined you lived 120 days in NJ, you can say you lived there from Jan 1 to April 30. Then in NY from May 1 to Dec 31. This is just an example how you would allocate even though your situation is different.
This may not be correct but you are still allocating the correct amount of days that you lived in each state.