I had a problem with my sticky eyes again, but I think it is all working now.
The problem was rotating my character caused the sticky eye blendshapes to turn on, as the eyes I was using for my set driven keys moved with the orient constraint. This meant that if my character was turned sideways, but looking ahead of itself, the sticky eyes behaved as though the character was looking sideways.
The problem is more obvious in this image, through the Face_Cam and with the eyelashes hidden - the look-left sticky eyes are enabled here.
I was able to resolve this by grouping the eyes from the rig again and using an orient constraint to turn them with my character's head.
The eyes turn with my character now, so the rotation values are on this new group, rather than the joints that are driving my blendshapes. I have played around with this setup and think it has fixed the problem without causing any new ones.