I have four kids (now young adults) and we went when they were younger and money was tighter.
It sounds like you will be staying with family and have no need for a hotel?
Here is a link to mousesavers which may give you some ideas
www.mousesavers.com. Sign up for their newsletter and you can find even better prices. Not hugely better but better. Maybe save some on park tickets.
If your biggest expense is food (after tickets) then that is where to focus. Eat breakfast at home or in the SUV. Fruit. Bagels. Granolas. Etc.
Bring lunch / sandwiches. Put them in a DL or DCA locker along with some ice to keep things cool. $11. They will not let you bring coolers in through security but sandwiches and chips are OK albeit technically against the rules. You have the option of leaving food in your car but that will burn a lot of time getting back and forth to the car. So use a locker for $11.
Eat dinner at one of the many good and (relatively) inexpensive places at DLR. At DCA I would suggest Flo's V-8 Diner. Eight people will cost you $150 maybe less. At DL do Plaza Inn or Redd Rockett's Pizza Port. Another $150 maybe less. So $300 for food plus what you make or bring from home. That will be maybe another $100 for breakfast and sandwiches from home.
June 15 - A day trip to DCA. This is a Monday.
Visit DCA on a Saturday, Tuesday or Thursday. Those days DCA does NOT have Extra Magic Hours in the morning (DL does) and you will be on equal footing with all other guests. Do not visit DCA on Monday, June 15.
June 17 - A day trip to DL. This is a Wednesday.
Visit DL on a Sunday, Monday, Wednesday or Friday. Those days DL does NOT have Extra Magic Hours in the morning (DCA does) and you will be on equal footing with all other guests. Visiting DL on Wednesday, June 17 is fine in this regard. However, read on.
For more info see "Early Admission (MM/EMH/APEE) Super Thread" by DLR29
www.disboards.com/showthread.php?t=1675403
Grad nights. These will impact DL and DCA.
See "The Scoop on Grad Nights at DLR in May and June"
www.disboards.com/showthread.php?t=2298400.
June 17 is a grad night. See previous link.
I would not visit DL or DCA June 17 if you can change it.
If I were you I would visit DCA on June 16 Tuesday (no grad night and no DCA EMH) and DL on June 19 Friday (no grad night and no DL EMH). All else being equal.
Have a great trip with those kids!