Stay away from Pershing, go to Long Branch instead!
When I stayed at Pershing it felt like an RV park. There were very few tent sites and the bathrooms were not very clean. There is a bad odor coming from the pit toilets like they have not been pumped all summer, the odor permeated through the back part of the campground.
It does have showers and an amphitheater with a few trails and some play ground equipment. On a positive note the grounds were mowed and clean.
Honestly, Long Branch State Park is 40 miles away and is a 5 star park.