Are you positive the shake is coming from the rear end? Those symptoms sure sound like there's excessive play in the front end. Worn balljoints or tierods or bushings are prime suspects. I'd see if a shop that works on Step Vans which are similar to your chassis would check it out for you. The "bell-cranks" on that chassis are notorious for loose, sloppy fits.
https://www.amazon.com/SuperSteer-P3032-Bell-Crank/dp/B009WQQ64M These bell-cranks are expensive but are a positive fix to get that play out. GM should be forced to replace them on every "P" chassis ever made.