I think you'd really like 20816.
We were a lot like your family in 2003-04, and we moved to 20816, and our quality of life went up pretty dramatically. A lot of people in your stage of life make the move to the close-in suburbs and never look back.
This zip code borders is the area right near Western and Massachusetts Ave. So it borders DC.
It's very close in, and yet is much more kid-friendly than DC. It feels like you're way out in the suburbs, when you are still within 1 mile of Metro. (My DH still walks or bikes to Metro most days.)
In the past on this forum, when I have suggested 20816, a few people seem very opposed to the idea of moving to the close-in suburbs, and claim that you'll never be able to walk anywhere and that your life will be an empty suburban nightmare. But IMHO, those people have not given the close-in suburbs a try. It's a pretty nice existence, with a lot more amenities than DC. The people are nice and respectful neighbors, and they take care of their homes. We have a neighbor on our street who moved from the area near Catholic University, and she still cannot believe how quiet the neighborhood is, and how rarely she hears sirens anymore.
This place is pending, but you get the idea --
https://www.realtor.com/realestateandhomes-detail/5221-Duvall-Dr_Bethesda_MD_20816_M61897-13081?from=srp-list-card