Yes. Because your employer withheld Virginia income tax by mistake, you will get back your entire withholding (W-2 Box 17).
You will have to make a separate payment to DC. Withholding to VA can't be transferred to DC. Money has to come to you and then be sent to DC.
Your VA refund will probably be less than the DC tax because the DC tax rate is higher.
It's possible you can get your VA refund before the tax deadline (April 18 this year) and use that to pay DC.
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