Well, we (terrebandit & I) finally dug into the steering head bearings... sorta. Dropped the bars and did the Suzuki factory "initial force" steering adjustment (page 5-27) test. Could not get any kind of reading on the scale it was so smooth and "loose". Not loose like wobbly just no resistance to bar movement.
I used the smallest ball peen hammer and a drift punch to tighten the steering stem nut... it moved very, very easily over a 1/4 rotation. We kept checking the tension and it was finally starting to register something on the scale. Kept on tightening until it was in spec (high end of the 7 - 17 oz. range) and yet the steering head/fork assembly was still smooth with no notchiness noted.
Torqued down the steering stem head nut and put it all together. A final measurement showed the tension got tighter (and now really on the high end) but we decided to leave it. I also noted that it had more tension one direction than the other - maybe cable routing caused that, not sure. Next time I'll not get it quite so tight to compensate for the final torquing of the head nut.
Took it out for a little 100 mile romp and WOW... what a huge difference in handling. Practically all of the headshake is gone. Several tests were done in the 40 - 55 mph range and I was able to generate headshake in only one condition. Smooth asphalt headed downhill leaning back slightly caused it twice. The rest of the time it was rock solid. There was overall improvement in the bike's handling at other speeds as well. Everything felt more stable - one has to presume it's similar to installing a steering dampener.
I will put a few hundred more miles on it and recheck the tension. I may have gone slightly too tight but things are still very smooth. In reading the factory manual it says to torque the steering stem nut to a specified setting and then back off 1/4 turn. I would think that will leave the nut fairly tight still which should be tighter than I first found mine. We were both surprised how easily that nut moved at first.
Now add to the mix a new 009 Shinko rear tire and things really feel different. I'm liking the Shinko tire - very grippy feeling and of course with no squared off edges the steering is a LOT quicker feeling. We also did a valve adjustment - that required no adjustments (yippee)!!!! I'm all set for a few months of riding!!!!