WHDLOAD DAMNIT!
I am glad you are also experiencing this problem, dougal. I had a feeling all my Amiga hardware was failing, but I get the exact same results. Games that aren't using WHDLoad work fine with CD32 and 2-button controls, but they're all failing with WHDLoad. They never used to, it's just been in the last version or two that I've experienced this problem on my Amigas.
I wish we could report it but they probably won't believe us that there's a bug.