If you have an F-53 Chassis, there was a problem with the mounting hole for the rim being out of round. Since you have 22" tires, that may not be you.
If the shimmy you described just showed up with the new tires you just had installed, then the problem is with the balance, since it only occurs at specific speeds. A typical symptom of tires out of balance.
I have no idea what weight is acceptable. 12 oz seems a bit high to me. indicating something is not kosher with your tire.
I know this is hind sight, but the Goodyear tires would not be my first choice. I had them on my rig. I had nothing but problems with them. 2 Blowouts, and constant balance problems.