Yes take the shuttle!! Traffic is not fun!!
The shuttle leaves at 9am and drops you off at the capitol at 10am. Pick-up is at 5pm so that you are back at the campground at 6pm. That gives you 7 hours to explore DC. Shuttle costs 14 USD per adult (as per April 2018). Very convenient service!
It's pretty far. It's absolutely doable but it's about 45 to an hour away on the train, the traffic is horrible and parking near/ around the mall is tough. They have shuttles but the last time to be picked up is early (between 5pm to 8pm I cant remember the exact time).
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