Psychlist, thanks for your comments and observations. I occasionally carry a passenger, but not very often.
I will start with upping the pre-load - at 13 st, I'm not hugely heavy but it's possibly a factor. Comp (and rebound) damping adjustments will follow if required. The back end feels like it's flailing a little, so stiffening it is probably the thing to do.
Glyn, thanks too for your suggestions. Tyre pressure is always checked on my own pump, for consistency's sake. For some reason, I can always feel when I'm riding with under-inflated tyres, even if it's only a couple of psi! As for the head bearing, my garage told me at that last service that there was a suggestion of a slight notch when the bars are pointed straight-ahead, but there was no movement. I asked them whether the bearing needed replacing but they said it could wait. Trouble is, once you hear something like that, you start imagining problems there!
So, I will be doing some incremental changes on the rear suspension - if that doesn't eliminate the very slight nervousness my 750 seems to have, I'll look at the head bearings (and maybe fork oil, too).
Cheers!