For your first question, No, you do not need to file both states if you did not earn any income in WV. There would be nothing to report for WV. However, if you were receiving unemployment benefits while living in WV, you would need to file a WV part-year return. Even though unemployment insurance is paid from NJ, it would be taxable in WV if you were living there when you received it. But if you had no income while living in WV, there is nothing to tax there.
It depends, however, on the second part of the question. Make sure you are filing the correct return. If you began the NJ part-year return, and entered the correct move dates, then I would leave this information as is. If you are not in the NJ part-year return, then the system may be presuming a NJ address for a "NJ resident" (which you no longer are). I would switch to the part-year return in this case.
I don't think that this it's going to have a major impact on your return if you leave the NJ information on the screen in question. In part I suspect that they need a physical address to determine the correct municipality code, and a WV address will not provide that information to NJ. By listing the NJ address you were living at during the time you were a NJ resident, you will be providing the correct information. Always keep documentation with questions like these, in case the state were to ask questions later on.
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