It shakes pretty violently at random times below 45mph and always above it. Trying to go above 45mph feels unsafe, to put it lightly.
The tires are radials, but could they have still flat spotted and/or gotten out of balance enough to cause this? Is there a cure? Any other thoughts besides tires sitting too long?
Yeah, a year on the truck would have been nice to see. I just figured you had an old school Suburban, since that's how most people roll down there.
Seriously though, check the rear u-bolts. The tires would be the last thing I'd suspect. I'd check all the A-arm bushings, torsion bars and whatever else the front of that thing has on it to make sure nothing is worn out. Lastly, I'd take it to a tire shop get the tires balanced and the truck aligned.