I'd like to meet a man who can shoot accurately on a moving horse. I think they should be able to shoot on horseback, but significantly worse when moving. While off of the horse, they should be much more accurate, but it should really come down to the playstyles of the group. Perhaps some would want to quickly flank around the enemy, dismount, get excellent shots off, and then upon getting charged, mount and disperse. Others may prefer being a shock unit rather than a skirmish unit, by picking out a blobbed, unprepared target, getting in their proximity, volleying, and then charging with sabers, which is much harder to pull off, but should still be allowed. This way, cavalry regiments that can successfully pull off both styles of fighting, and perhaps even more, will have extreme advantage on the field.
As far as fairness in events go, I'd say that if a cavalry regiment runs up, volleys, and doesn't charge, that should be punishable. What's really unfair is letting cavalry get fairly accurate, but very up close shots, but then letting them gallop away quickly.
The big solution I see though, to nerfing the cavalry, is slowing the horses a little. This way they'll be less willing to take risks, because it would become harder to run up to infantry, shoot on horseback at a halt, and run away the second someone gets near. Just a slight change in their speed could do the trick.