We live in the burbs of DC and go to the Smithsonian's all the time. We were just at National Gallery of Art for a kids film yesterday. Here is the problem with the Smithsonian's, they are so huge that you could spend just a day in each museum and not see everything. The American History museum opened a couple of years ago and is fabulous and your boys would love it. Air and Space downtown has a huge new Wright Brothers section that is amazing. My kids are too young to go but I've heard the Holocaust Museum is also really good. I've been meaning to do the Federal Reserve tour too, getting tickets is kind of tricky and I need to do that before the summer.
The memorials are also amazing for a history buff, your boys will LOVE the WWII memorial.
The Neuseum is not free but I think it is amazing and one of the best in DC. They have part of the antenna from the north tower with the headlines from around the world. I tear up right now thinking about it, very emotional. I'm not a fan of the Spy museum or Crime and Punishment, they were okay but not as high yield as the other museums.
National Geographic museum can also be very good, depending on the exhibit. They will probably be too far to walk to depending on where you are staying and it is a small museum.
Air and Space on the Mall is by far the most popular museum on the mall, the lines and crowds are tough in the summer, make that one of your first stops also, I'd get there before it opened if I was going to see it in the Summer. Last time I went in the summer the line was probably 200 people long. My kids love that museum.
I think the hop-on hop-off tours are totally worth the money. You can see so much of DC, and really get a good idea of where everything is located. You can get a two day tour and it really saves you from getting exhausted in the heat. Also they will take you by the Pentagon and through Georgetown which you might not get to see on your own. http://www.dctours.us. I'd focus on the memorials when I was on the tour bus because they are further out and most of the museums are on the mall.
Mount Vernon is owned by a private foundation and it is also really nice with lots of exhibits and films. The traffic to get there through Northern Virginia is ridiculous so I were doing it I would take a cab to SW DC and take this Spirit Cruise over, (haven't done this but it looks great) you will see many landmarks along the way and the Potomac is beautiful.
There is a National Harbor pickup also but Nat Harbor is far from downtown DC and is an artificial town... not a fan.
DC in August is hot and crowded, it's not uncommon to encounter long lines to enter the museums and there is alot of walking involved, it can be exhausting. School here does not go back in session until August 26.
National Archives, (see Constitution, Declaration of Indepence) always has a super long line, so make that your first stop one morning. Just looked on their web page, looks like you can now make a reservation!! http://www.archives.gov/nae/visit/reserved-visits.html
You can get away without a car in DC and I would totally recommend you do that, parking is usually 15 to 20 per day, even if you are staying in the hotels they charge for parking.
Assateague is a jewel and is close to Ocean City Maryland. There are two campgrounds on the island, State and National. I doubt you want to go camping and you have to book a site far in advance because they book up fast. Problem is that Ocean city is just what you want to avoid, noisy and commercial but the kids do love the boardwalk and beach. Bethany Beach, Dewey Beach, Rehoboth Beach are quiet and more family beaches but coming from Europe you may want more excitement and your goal sounds like history.
I have friends that just went to VB last summer and they loved it, I'll have to ask where they stayed because they found it very laid back??? Maybe after living in DC anything is laid back! Another poster mentioned the battleship and that sounds wonderful for your boys. I think battleships are some of my favorite military tours and so interesting.
The National Park Service does tours of the battlefields and they are very good, but I'm not a huge fan of battlefields, but then again I'm not a history buff, and the drive to Gettysburg from DC is 2 hours one way without traffic. That being said the NPS tour is excellent!
Another place outside of DC that I think is a must see for your family is Udvar Hazy... My H is military and I have been to dozens of aviation museums, subs, battleships, boats, etc. Bar none, best aviation museum in the states. You would either have to rent a car or cab it. Parking is I think $15 dollars at Udvar Hazy. If you drive over, Manassas Battleground is not far.
Udvar Hazy has a space shuttle, flight simulators, cool engine displays (which are really informative), tons of planes, rockets and IMAX films...
You know... it all depends on how you like to travel, I hate crowds, traffic and like down time, so I'd probably stay in DC the whole time, and make a trip over to Mount Vernon for a day, Udvar Hazy for a day and just really explore the city. But seeing that you did Boston and NYC in one trip it sounds like you are more tenacious and stronger than I am.
NPS memorial tours
I like this page to find events quickly...
Smithsonian has fantastic family events so check their web page before you go, or ask at the information desk whats going on that day. You get a 10% discount on shopping and food and reduced rates on IMAX if you are a Smithsonian member, membership is only $20 so you make back the money on discounted Imax at Air and Space. Oh and Air and Space is the most popular museum on the mall, get there the day before it opens because the summer crowds are tough.
I sound like a big baby,
I would stay in the Penn Quarter section, good restaurants and you are close to the museums and sites. I'm not a big fan of staying further out and metroing in, you have to buy tickets, it's crowded in the summer and stops every two feet in DC and Maryland suburbs so it can take more time. I would metro from Penn Quarter/Verizon Center to someplace like Dupont Circle if you wanted to do a walking tour of the embassies or an african american heritage tour.
Check out walking tours and heritage tours DC on this page
Outside of DC...
Williamsburg is run by a foundation and I think they have done a great job in recreating and preserving the town, there are craftsmen making furniture, bricks... etc... We stayed there a couple of years ago and really enjoyed it. It was worth it to get a room where we could walk right into Colonial Williamsburg and we did a ghost tour which was really fun, the actors were great. We also went in August and got stuck in Virginia and weekend beach traffic, the trip ended up taking almost 5 1/2 hours from DC.
Don't know if you have a smartphone, but Yelp is used heavily in this area and it can help you find better restaurants and more reasonably priced places to eat. DC is expensive.
I also really love Baltimore for the science museum and Aquarium and they have some historical boats you can tour. There is a railroad museum there too.
Feel free to PM me if I can be of any help.