If the vibration is in fact starting from start and getting worse when sped up then you have a tire issue. Swapping the tires with the rears will at least change the vibration from front (steering wheel) to rear (seat of the pants feeling)

All it takes is a bounce of a curb and a belt shifts and its junk. With all the pot holes this winter here in CT, I am surprised my 3500 doesn't have the same issue. If your tires are worn out then replacement is necessary. Swap your spare into the mix (try swapping out the left then the right with the spare) and see if that helps.

I aligned vehicles for GM for a lot of years and that same type of issue was common with all larger trucks (2500 and 3500) because of E-ply tires. Once the tires are worn down they heat up and a good bump would send them in a bad way.

Hope this helped!
Good luck and keep us informed.
Joe