I don't have the 20 years of life experience that others have on this question, but I thought I'd chime in.
I love DC, but at this time I do not want to live there. My cost of living would be ~3x what it is right now in Madison, WI. I know, because I have lived there. Madison even has a cost of living above American average. A cost of living calculator would not tell you this, but living in DC results in necessary lifestyle inflation. Bah, other crusty Mustachians will say, you must go for Maximum Mustache! You will eat at more restaurants, because there are more of them. You will go to more entertainment events, such as concerts, because there are more of them. Even if you tighten the spigot to 1% of what DC has to offer, the 1% is much, much more than you can access in other states.
I used to live next to the Franconia-Springfield station and work in Reston, 2 blocks from the new Wiehle station. I really wished that they would open the Silver Line, as my commute was about an hour driving. The traffic is abysmal.
Commuting is correlated with a huge drop in happiness, and you're best off keeping it 15 minutes or less. It can also be costly and time consuming.
http://www.mrmoneymustache.com/2011/10/06/the-true-cost-of-commuting/You sound like you have a great situation in Utah, and YMMV, but if I were in your shoes, I'd stay put. Lots of family and friends in your current location and grad school opportunities... You do not know what it is like to fight Beltway traffic twice a day, and you may never have to learn.
At minimum, it might be nice for your family to test drive living there for a week while staying in an Airbnb in a real residential neighborhood. (A week of only you and your husband scoping out potential living spaces would work, too.) People can argue over whether it's a good idea to move into the DC area, but you are the only one who can actually make a choice for your family. Long-term DC residents on this thread might scream that a week is far too short, but it'll give you at least a little bit of an idea of what it would be like to live there. Have your husband commute to and from the new office at peak rush hour. If you are planning on using public transit, there's a handy Tyson's Corner station on the Silver Line for him.
http://www.wjla.com/articles/2014/07/after-fanfare-long-awaited-silver-line-debuts-shortly-after-noon-105469.html Take your kids into DC to see the museums. Shop at the grocery store that you'd use if you lived there. Get a feel for it. It would be a nasty surprise if you uprooted yourselves only to find that it's not a good fit. Conversely, you might find that the shoe fits. Only you can decide.
The amenities are fantastic, and I love DC's museums. Your son especially would be in a great environment for his interests, which frankly sound a lot like mine.