We have stayed overnight at Sinks Canyon twice, June 26, 2009 and May 27, 2014. Not sure what day of week those were, but we got a site without a reservation (didn't know they take reservations). I doubt you have a problem getting a site the second week of May.
The road to the campground is paved, as are the campsites. It's about 7 miles west of Lander and about 3/4 miles further into the canyon from the Sinks visitor center. There's a pedestrian suspension bridge over the river at the campground to some trails and a parking area further into the canyon for a trail to a waterfall.
Google Earth image of Sinks Canyon Campground
2003 Winnebago Minnie 24F - Ford E-450๐