Maybe one rear shock is dampening and the other isn't? Are the spring tension adjustments the same on each rear shock?

Have you tried swapping the front wheels between the two 360's? If this problem hasn't been fixed yet I would suspect a dodgy front tire. Rare to see that but it does happen.

While most of the comments so far seem focused on the front end and ensuring both the swingarm and headstock are correctly tightened, as well as looking at the tire pressures and front wheel, may I also suggest looking at the rear?

Since you apparently have 2 of the same bikes, and this one is acting differently, you do have some components you can swap around to prove/disprove any theories. Certainly swapping the entire front wheel will be quick and hopefully show you some improvement (??).

If at a loss, try also swapping the rears. A bit more work, yes. But you seem very well mechanically inclined so no big issue. A looong time ago an instructor pointed out to us that the rear wheel can have a distinct affect on how the front end felt, because that is the end that gets the power applied to. If the tire is not well seated, under/over inflated, the spokes loose, or not trued good enough, power applied to the wheel will amplify those flaws.

I note you rebuilt the wheels, would it be worth your time to recheck them now after you have ridden the bike a bit, to see if the side-to-side play and 'hop' have changed since they were initially trued?

Plus, if you did swap the "good" wheel/tire sets with the "bad" sets, and you have no change in behavior, that also tells you something? Just a thought.

Swapping parts from the bike that handles well sounds like a very good idea! I would start with the front end.

Hello everyone, I can now say the bike is fixed.
I believe what fixed it was replacing the headset bearings.
also after replacing the headset bearing I used a string alignment method and adjusted the front wheel over 1/16" very minor.
I am using an allballs tapered headset bearing so I cannot give advice on the proper torque for the OEM ball bearing.

I ended up at 22 foot pounds of preload on the tapered headset.
I also ended up at 25 psi of air pressure in my shinko trials tires for the road. 10-15 off-road.

I have ridden the bike extensively now at low speeds around town and the handling is good at low speeds.
I have also done quite a few miles at high speed 80MPH+ over some rough roads without any tank slappers or speedwobble.

Thanks for the input everyone!! 

VERY interesting thanks. But not sure why you need 'any' preload on the head bearings. Is it tapers top & bottom?. 22 would be enought to put dents in the top race if that's a ball bearing type. Probably wouldn't hurt taper rollers & just act like a steering damper--well my cheap version of a steering damper.

Sounds like it RT
22lbs on an old wheel bearing design?

The wheel bearings were to torque nut while spinning spindle to seat.
Loosen and hand tighten to nearest notch on castle nut for the cotter pin.

cliff

Taper roller head bearings, tightened down to 22ft lbs, will mean the original problem is back quite soon.

I take it you meant you tightened the top yoke bolt to 22 ft lbs and not the adjuster ring nut?

If you tightened the ring nut to 22ft lbs it will wear the bearings rapidly. Only time you generally apply a significant torque to any thrust bearing holding nut is where the preload is controlled by a spacer and or shim arrangement.
(apart from a certain Kessler axle from a 200 tonne dump truck.  Wheelbearing nut preload torque for new bearings is 1000nm, but then again those bearings are well over a foot in diameter.  You need a scaffold pole to turn the hub afterwards.  That is the only time I ever saw a direct torque preload applied to a taper bearing).

Otherwise you just need to take out all free play without applying significant preload.

The very poor yamaha system of adjustment means significant trial and error by tightening the ring nut a little, then tightening the upper yoke bolt to take out the remaining slop.
If it still has slop, undo the yoke bolt, tighten the ring nut a fraction and try again until the slop has only just gone.

To add, wheel bearings have a spacer between so you can tighten the axle as tight as you like--till ya strip it--but with head bearings ya trying to draw them together, well sort off, as no distance piece.
Reminds me of my Bultaco Pursang Mk8 1975 that had ordinary steering head bearings as in normal bearings like big wheel bearings. Worked good too.
I guess ya can't come to much harm though if its behaving itself.
Please update us to see how it last like that. Unless as someone said--are you just talking torquing the top nut, I guess so, as can't easily torque the adjuster ring. I think maybe i misunderstood what you meant now i think about it. Strange thing is then, 'if that', why did tightening the nut in that way fix the wandering. Don't worry i'm tired & woffling.