I think maybe I've got a combination of problems, then!

I'm winding by hand - it's quite probable that I'm not doing it tightly enough. I do have a drill, and I don't know why it never occurred to me to use it (but my jewellery teacher taught us doing it by hand!). Regarding the ratio of mandrel to gauge, is there some kind of formula you can point me to? But tbh, it happens when I'm making smallish rings (~4mm) with not-that thin wire (0.8 or 1mm), so is it more likely that I'm not winding tightly enough, do you think?

snow_imp - when you say 'pushing against the coil', do you mean that you reverse the saw blade so that it cuts on the up-stroke? Or something else? I'm a bit dozy today..

Thanks again for the pointers - I'm very grateful.