Stabilizers are nice, and can calm headshake, but you don't need one yet.
They customize the effort to move the handlebars, which is nice, but they won't prevent a real tank slapper...and, they can bandaid other problems...which you have.
That bike is capable of being glass smooth all the way to top speed...without a steering dampner.
So, fix the problems first...Then buy a dampner, Scott's, GPR, Ohlins, several to chose from.
For your suspension, if it's never been serviced in 35k miles, or not in the last couple of years, then the fork oil is done, and needs changed, and the rear shock could use an oil change and recharge as well.
Then have the suspension Sag set for Your weight, including gear.
If you're over 200 lbs, you need heavier fork springs installed.
Also, look at the simple things, like tire wear and condition, as well as psi and proper balance.
Not everyone balances motorcycle tires(they should), and sometimes weights get slung off.
I have ridden on tires that felt great, but at about 110-120mph, the rear began to shake lightly, feel out of control.
I balanced it for him, and it was smooth to top speed.
I would check to make sure the wheels were not bent, or the rotors, as if bent, they will catch the pads in the high spot.
Check the wheel bearings, steering neck bearings, and swingarm bearings for play.
Check for correct chain tension, as too tight limits rear suspension travel, and front and rear suspension work together, so if one is incorrect, or restricted, it effects the other, and throws off the suspension as a whole.
I would also check for any loose or missing body panels, as they can upset aerodynamics at speed.
