Much good tire/damper info above... FM doesn't disappoint.
The new tire will likely restore order in your world.
In answer to one of your other questions:
q: Can a steering damper mask other poor handling problems?
a: Yes.
It is worth checking some stuff that is pretty easy:
1. Slop in the steering head bearings. While you have the front wheel off the ground for the tire swap, grab the wheel and with it pointing straight ahead, push and pull it forward and backwards. There should be no slop or clunking. If there is, maybe steering stem bearings are worn. Also while the front wheel is off the ground, turn the bars right and left across the point where the wheel faces forward. If you can feel a "notch" at the straight ahead point, there is some wear that is no helping anything.
2. Spring sag. If either the front or rear springs sag more than about 35% while loaded with rider and stuff you carry, especially the rear, then maybe more preload, or possibly new, stronger springs will help. I find that if I can put a digital angle finder (or a bubble level) on a bottom frame rail, and get it level, head shake tendencies are reduced a lot.
3. Bike swing arm bearing slop. This should not be an issue on a GW, but worth checking. Rear wheel off the ground, grab back of tire, pull and push left to right of bike. Should not be any noticeable clunking or movement. If there is, probably time to look at bearing, or at least bearing preload.
3a. Sidecar swing arm bushings. Same test as bike swingarm, raise up until tire is off the ground, push/pull rear of tire left to right and back to look for slop/clunks.
Realistically, the front tire change will likely be so much better that you will stop looking, but the above stuff should be checked once in a while.
I run a hydraulic click adjust damper on my steering stem from MSC moto.
They are stupid expensive, but I find that I like it off road. I dial it in pretty hard for rocks and sand; leave it almost off for everything else. The automotive one that you are looking to replace should be all you want/need for street driveing. Goal should be to to get things to where you don't "need" the damper, but just like it!
Good luck!
