replacing the cassette will not fix the problem unless the lack of engagement is between the cassette teeth and the chain, which it sounds like it is not.
there are pawls between the freehub body (onto which the cassette mounts) and the hub body (onto which the freehub body mounts) which provide the ratchet action that you are apparently missing.
there are several options for problems. if the hub has been recently serviced, over greasing (or too thick grease) in the pawl area will prevent them from rising up enough to engage the freehub body serrations. alternately, something is worn. possibly the springs that push the pawls up, possibly the freehub serrations that they engage, possible the pawls themselves. only way to know is to disassemble.
the bigger problem (i suspect) will be finding replacement parts for your long ago discontinued hub, especially as suntour no longer exists! watch ebay for a similar one, or rebuild the wheel around another type (which might require other drivetrain changes). fun!
Could anyone guess a date for this pretty old Edison cyclo-cross machine?