Soooo, the back story on my shake is detailed in my posts above.
In the process of trying to have my wheels and tires properly balanced I learned that my Dayton wire wheels, with about only 1,000 miles on them, were quite out of round. One was 100/1000ths out. So I bit the bullet. I ordered four new tires, (my old tires were more than 10 years old) sent the wheels to Hendrix and gave him the green light. He tuned the wheels, mounted the new tires, shaved them, balanced the assemblies and sent then back to me. They arrived today. My god what a diference. For the first time since I put my car back on the road, it drives smooth and shake free at all speeds up to 80 mph (in rush hour traffic I didn't get a chance to go any faster).
FWIW, I replaced my 185 70/15 Vrederstein Sports Classics with 165 80/15 Vredersteins. This also made a huge difference. The 165s are much easier to turn in the driveway and at parking lot speeds, plus, to my surprise, the car accelerates faster, responds faster in turns, the "dead spot" in the steering box is much smaller and the brakes feel like they grab faster. While I'm sure the car will let go a bit sooner at the limits, it feels the way it is supposed to feel. I had no idea that this change was going to make so much difference. If you are wondering what size/model tires to get for your BJ8, I strongly urge you to buy the 165/15 Sports Classics. The tire-fender gap isn't much different either.
This was an expensive project. About $650 for the new tires and $880 all in for the work, new inner tubes and shipping to and from Hendrix. But the benefits exceed the cost. I hope the weather gives me some opportunity to drive it before the season is over.