<<If you did not work or live in N J, then you should be having tax withheld for DC. Then you will file a tax return for your non-resident state first (DC) for 2019 and a return for VA next.>>
No - there is no non-resident DC return. If you work in DC and live in any other state, you're not subject to DC income tax.
Have you literally moved to VA and it is now your home (your domicile in tax terminology)? If so, your DC employer should be withholding VA taxes. Income earned in DC by a resident of VA is taxable only by VA. Only DC residents are subject to DC's income tax.
But if your own main home (your domicile) is still in NJ, and you are only in VA temporarily and for fewer than 183 days of the tax year - in other words if you haven't really "moved" to VA, then for tax purposes you remain a NJ resident and your DC employer should withhold NJ taxes.
But be aware that a person who lives in Virginia, or maintains a place of abode here, for more than 183 days during the year, or who is a legal (domiciliary) resident of the Commonwealth, is considered a Virginia resident for income tax purposes.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
