My machine (admittedly not quite so old) sometimes leaves the washing dripping wet despite having been set to spin at 1200 rpm. It is often obviously because the load was off kilter. It has a "just spin" cycle which I then run, after jiggling the contents about a bit, and then all is fine.
As your machine does still spin some of the time, is therefore able to do so, can you check the manual to see if it has a programme which just spins? Sorry, though, if that's a rude question, Helterskelter as you are clearly au fait with the mechanics and have probably thought of that already.