Get a pair of 'slick-ish' tires for the MTB and go for the road.

I have done my fair share of sidewalk riding even though it is illegal, and I have to say that you are at the very least responsible for the pedestrians, and making sure that they don't get hit by you. This can very much mean that you'd have to get off the bike and walk with it, which would defeat the reason for having the bike.

What I recommend is stick to the streets as long as you can and you certainly need to plan a new route through less busy streets. Do not think that you'd be able to do a straight shoot route as if you were driving. Since you are new to the area, I recommend at the very least to 'scout' the streets with google maps, and even better if you can go out yourself to take a look around.

Rear view mirrors are great too, but you do need a certain level of nerves to ride along with cars. If you are all over the place and can't hold your wheel straight, sidewalk is a consideration. If you can hold the wheel straight without a problem, you should be good to go. Remember that cars are generally not out to hit you, and will try to avoid you.

But that is my experience from across the coast. I am sure there are a few DC commuters that will chime in.