The way things should work when it comes to the different state DMV agencies and the way that they actually do work are often quite different. It might be a good idea to just apply for a new license in DC. Good luck.
I agree with Mr. Nielsen.
The Interstate Compact signed by the various states (and DC) should mean that Virginia's suspension of your license will cause DC to refuse to grant you a license. But bureaucracies being what they are, you may want to apply anyway.
But don't drive in Virginia!