I would like to just let all of you Silverado 1500 owners know that I to have had a 2015 that was shaking right away when I left the dealer. I soon found this thread after but it was to late for me.
I fought with GM and the deal over this for more than 3 months and I finally got them to give me "trade assist" to get out of the 2015 that was shaking so bad that you would wear your coffee if it was in the console.
GM came up with $6500 in trade assist to get me to where my payments were for the `15 with 2500 miles that I was having issues with shaking. I ended up test driving 3 2016 trucks before I found one that didn't shake going down the road at the magic number of 74 mph.
If you are buying a new 1500 take it for a test drive on the highway! if it shakes at 75 don't even bother buying that truck, ask to take another for a ride.
It to me seems really random and honestly it seems like it is something in the transmission causing this. From me taking all these trucks for rides the last one that doesn't shake shifts totally different then the rest.
Good Luck all.. Don't delay if yours shakes, it wont go away, keep hammering on the dealer and tell them you want to get out of it and they need to come up with some money to help you.