I have been upstream to Raquette Falls several times and never had a problem with muddy campsites.
Here is a good map:
Raquette River below the Falls
And therein lies a secret: there is a well hidden, very large, overflow campsite at Raquette Falls, so you don't have to worry about all the spots being taken. To find the overflow site, look for path just before you get to the beach (which is the takeout before the falls). On the map, the overflow site is the one to the right of the other two campsite and the lean-to, near the Falls. There is also a NYS Interior Caretaker (like a forest ranger) stationed there, with a cabin nearby but not shown on the map.
And by the way, all the campsites are on the left if you are going upstream. The right shore belongs to the Nature Conservancy and includes the very famous Follensby Pond (known for the Philosopher's Camp long ago). It is prominent on the map linked above but is not labelled for some reason (i.e. they don't want you going there!). The Nature Conservancy is trying to figure out how to protect the pond while giving the state an easement for the land.
And I forgot the mention: The Crusher is NOT a terrifying series of rapids! It is a pull-off where a rock crusher was located when the road was built, and later re-purposed as a boat launch.
I highly recommend this trip. Just allow enough time to get all the way to the Falls, in case the lean-tos on the way are taken. But if you see a site you like, you can grab it and then make the Falls a day trip the next day. Done that too!