Hope the new tapered steering head bearings completely resolve the handling stability issues . . . but if not, all the previous suggestions are excellent IMO, and here are yet some more possibilities perhaps worthy of consideration.
(1) Verify rear wheel alignment. Don't place blind faith in the chain adjuster alignment marks. Sight down the chain (eyeball chain for straightness from top of rear sprocket to front sprocket making sure the chain goes straight onto the sprockets without making any sideways turns).
(2) Verify alignment of front wheel with rear wheel. When the front wheel is pointing perfectly straight forward, the rear wheel should also be pointing perfectly straight forward. May use a long straight edge or string (sight down it) to assure this. When coming off sides of the rear tire (from rear end of rear tire) toward the front and on past the front wheel, the straight edge or straight string should be exactly the same slight distance from both sides of the front tire. The straight edge or string won't be up against both the sides of the front tire because of the smaller width tire on the front. If both tires were the same width, the straight edge or string would supposedly be fitting smoothly up against the rear tire sides and front tire sides. (Does this make sense?).
The objective is to assure that both front and rear wheels are perfectly in line with each other and both pointing straight ahead and in line with the bike when motoring straight down the road. Should be able to remove hands from the handlebars and easily steer it with only gentle knee pressure indefinitely along a straight road and with the bike remaining perfectly upright without any sideways lean and without any tendancy to pull one way or another as you motor along.
(3) Loose swing arm bolt (already mentioned) or worn swing arm bearings (bushings in earlier models) may also cause an unstable ride. On centerstand (rear wheel in the air), try to move the rear tire from side and look and listen for any play in the swing arm between the swing arm and the frame.
(4) Confirm both rear shocks adjusted to same preload, and that neither shock is leaking.
(5) Assure forks are equally adjusted in their clamps and are not binding; front wheel fork caps are properly installed and tightened; correct amount and viscosity of fork oil.
(6) As already mentioned -- Assure front wheel bearings are not worn out (or rear wheel bearings).
(7) Has bike ever been dropped? Could result in triple tree (steering head) damage which is sometimes difficult to see by visual examination. Swing arm bent or twisted? Frame bent? Fork bent?
(8) tires balanced? either wheel bent or warped?
FWIW, I get new mail order tires mounted and balanced at the local Kaw dealer for $25 each wheel and consider it money well spent. And would pay the $35 - $50 charged by other places if necessary. Just hate, hate, hate personally mounting new tires
.