Denny: I probably cannot add much more insight that what has been given so far. But, have you considered that maybe the problem is not with the wheels and tires, but with the steering and suspension? Take for example my 1998 BMW 740iL. Love the car, but shimmy at speed is a constant battle with this car. Tire/wheel balance is important to the equation, but the problem lies with BMW's suspension design and related bushings. In order to keep the car without shimmy, the bushing must be replaced about every 25k miles. So, maybe you have a harmonic oscillation issue. The fact that slowing down cures the problem only to re-occur at a different speed leads me to believe the problem is in the suspension. Maybe change you toe-in setting slightly (i.e more toe-in). What is the condition of the tie-rod ends? Have you considered ensuring the wheels are balance both inside and out?
Regardless, I hope you get this figured out. I am happy to report that my '52 does not have this issue. All new steering components, rebuilt steering box, NEW wheels from Wheelsmith, new radials, new springs, new king pins. Toe-in set by me with a tape measure.
Oh, one more thing: You might consider that your brake drum(s) is (are) wildly out of balance. Maybe try a shop that can balance the wheel/tires mounted on the hub.
Last edited by DavidF; 05/20/2015 4:20 PM.