Solved the problem!
Actually, I got a very good response from Iomega. I e-mailed them the problem and within four hours got a response (on a Sunday!). However, I didn't get a chance to try their solution until yesterday.
It seems to have been cause by a conflict between the Iomega software and Microsoft's Intellipoint trackball software (Iomega having contacted Microsoft). The solution was to un-install Intellipoint, install the Iomega software and then re-install Intellipoint. It took about five minutes to do and everything works fine now.
Phew - end of crisis!
I wouldn't mind a CD-RW but I've only had the Zip a few months so need to keep using it for a bit to justify the cost. I'm not sure that I actually could fit a CD-RW on my antique machine, that would really overload the single parallel port! (USB? what's that?)
In any case, thanks for the suggestions everyone.
GG