It depends. Normally, you can't do three part-year returns in TurboTax. In your case, however, you only need to file two state returns: VA and NJ (Washington does not have a state income tax). For the system, you can simply tell the program that you lived in VA and NJ. Then, when you get to the NJ tax return, you will see a screen that asks about excluding income for your NJ return. Check on the boxes that are not NJ. As a NJ part-year resident, no income from the other states is taxable in NJ if you were not living there at the time.
Please see the following screenshot for assistance. (The state reflected in the shot is different, but the idea is the same):
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