Many many moons ago, we stayed at the Hilton and walked to the park one day. It wasn't too bad if you don't mind walking. The next day, we walked back to the hotel after a day at the park. They must have moved the hotel that day, because the walk back to the hotel was much longer.
I'd save your feet and take the shuttle. If you're ambitious you MIGHT be able to walk about 1/2 way and catch a tram from the smallish parking lot near DCA.