If I were going to do that (never)I would park my car and hitch to the base station, take the train up and hike down to 302. The mileage may not look long to you but the trail is quite rugged. It is easier to go from Washington across Monroe, Eisenhower, Pierce, Jackson and Webster than it is to go in reverse. Also, if you don't already have a ticket, you sometimes can't get one going down unless the cars are not full.
I have never stayed at Crawford campground but a couple of weeks ago I stayed at Ammonoosuc campground in Twin Mt. on Rt. 3. They had a bathhouse and laundry and they also had a shelter over the picnic table at each site.
As Andrew mentioned, it is often easier to ask people for a ride then it is to hitch.
_Sleeveless_