Yes. You will need to file a part-year return for both states since you moved during 2020.
Virginia and Washington DC have a reciprocity agreement. A reciprocal agreement, also called reciprocity, is an agreement between two states that allows residents of one state to request exemption from tax withholding in the other (reciprocal) state. This can save you the trouble of having to file multiple state returns.
Typically, If your employer has withheld taxes for Virginia instead of DC, you'll have to file for a refund from Virginia. You'll still file your resident return that also includes that income and pay tax on it. However, since you were a part-year resident of both states, you will need to file both returns as part-year resident.
Lastly, you will need to file a VA-4 Form and give it to your employer so only DC taxes will be taken out of your wages. You will need to re-certify your exemption every year.
I received a W-2 from my employer for both Virginia and DC. Because of the reciprocity agreement only Virginia had state taxes deducted. Do I still need to file DC as a state without state income taxes?
If you live in VA and work in DC, you would just have VA taxes taken out of your paycheck. You would only file a VA resident return. There is no need to file a DC return because of the reciprocity agreement between VA and DC