When we visited DC a few years ago (2013)... we flew in, but friends with a van drove in to play with us.
There are (at least for DC) find-parking websites that have maps (and frequently height-clearance info) of the lots around town.
Just Googling "parking washington dc" found the websites, many (most?) of which were independent of any "brand" of parking lot.
We managed to find locked underground parking with 9 feet of clearance (for one floor of the lot, the rest was 6.5') a few blocks inside of Q street, almost due north of the White House. It was priced competitively with "just plain cars" parking, and worked out well for the 5 days they were with us (they
slept in the AirB&B apartment we had).
If you'll fit under 9 (or was it 8'?) feet, it could be an option (i'll try to re-find the lot's address)
added: it was at roughly 1447 P Street NW, directly across from the Whole Foods.
In this google map street view, it's the "tunnel" directly behind the guy striding towards the right.
https://www.google.com/maps/@38.909...4!1spK-OsoAHMnFjWEMqJPfzBg!2e0!7i16384!8i8192
--dick
p.s. the websites also listed some locked surface lots, but two of them (when we visited) had been torn up for new building construction.