Welcome to the Forum John.
You could benefit from some suspension changes. My primary concern is that the shock is probably fairly compressed, which removes weight from the front wheel. With the OEM suspension the steering feel is already vague, and this nose up attitude makes it worse. So at a minimum I would re-spring the shock and forks.
The OEM spring rate for the shock is 12.5 kg/mm and it is a 7 x 2.25" coil (RaceTech). The RaceTech calculator suggests a spring rate of 17.18. That rate falls between available rates of 17.0 kg/mm and 17.8 kg/mm. I would choose the 17.0 spring.
The OEM spring rate for the forks is 0.77 kg/mm and the RaceTech calculator suggests 1.04 kg/mm. Once again, this suggested rate falls between available rates of 1.00 and 1.10 kg/mm. In my experience the RT calculator leans towards the too high a rate side, so I would suggest a rate of 1.00 kg/mm. If you choose a dual-rate spring, I would choose one with a starting rate no less than 1.00 kg/mm.
These two changes will help produce the correct free and race sag (putting the bike in balance), but will not improve the dynamics of the suspension. The forks on the FZ6 are under dampened, producing pronounced brake dive. However, they also tend to hydrolock during fast movements, resulting in a punishing ride. You can try a heavier weight fork oil, say 10w or 15w, which will improve fork dive, however the ride over harsh bumps will be worse. Damping rod type forks inherently have this worse-of-both-worlds characteristics. Gold Valve Emulators cure these problems by bypassing the damping rod for a rate-based valve.
There are no easy solutions for the shock. The OEM shock is an emulsion-type shock, and like the damping rod fork, it was chosen because it was inexpensive. Since the shock isn't rebuildable, the only solution is to replace it (the 17.0 kg/mm rate noted above would be correct for any replacement shock). The shock determines the ride comfort of the bike, and it's importance can not be over emphasized. There are some threads in the forums where members have modified shocks from other bikes to work on the FZ. I have done this in the past on other bikes and I was never happy with the result. The only plug and play solution that I'm aware of (that is currently in production) is a Ohlins shock. That is a $600 investment - but Ohlins will send it to you with the correct rate spring - and if you think about it that way, it's only a $450 investment.
So at the very least (and for your safety), respring the forks and shock. Depending on your budget and your commitment to the FZ as a bike you will keep, there are options to improve ride and handling.