Employer incorrectly paid taxes to VA even though I live/work in DC. Is it correct to file a nonresident form to VA? VA refund is much less than what I owe to DC.

So you neither live nor work in VA? What was your employer's thought process, or was this just a simple mistake?

You don't need to file a VA return if you neither lived nor worked in VA - EXCEPT in your case where the tax was withheld and sent to VA by mistake. 

Filing a non-resident return is the only way you can get that withholding back.

Of course, the difference is not just the tax rate, but all the additions and subtractions to federal income. If you want to look at this further, look at taxable income in both places and figure out why they are different. Also check the credits, which may be quite different.