Sounds like a worn out Rest-O-Ride seat.
Rest-O-Ride seats were once free for the asking just about everywhere but so many have been scrapped that they are not so readily available as they once were. Still, with patience, you should be able to find one but (just about) all will need to be rebuilt.
Properly rebuilding a Rest-O-Ride seat is time consuming and can be expensive but the end result is worth it if properly done.
A simple spring pan seat is the easy solution.