Honestly, I'd take a day or two off of NYC and add Philadelphia to it. Philly has nice historical places to visit, including the Liberty Bell and Independence Hall. It's an easy stop by bus or train between DC and NY.
I'm not that terribly impressed by NY. DC's good, though.

On a three-week trip, I don't really recommend driving across the country. Maybe fly into Salt Lake City and have a nice drive to Yellowstone National Park. Another destination near the middle is Denver, which also has pretty places close to it.
Or fly into Las Vegas, a fun destination in itself, that has nice national parks as well as the Grand Canyon nearby. Probably too hot in August, though.

Seattle is fun, and Portland is pretty easy to get to, 3 hours away, and you can take the beautiful drive around Columbia Gorge as mentioned above. Portland is also kind of a fun town to see.

I definitely wouldn't want to drive into San Francisco-way too many hills!

But LA-it's been a long time since I've been there, and, frankly, I'd look carefully at what you want to do there. I've heard it's overrated and boring.

Have fun on your trip!