This is proper Morgan Shimmy Vitruvius, as opposed to St.Malvern's dance (totally different thing!). Yes it's horrible! My old 4/4 I had in the early 80's suffered badly from it, in desperation I took it to the factory. The then chief test driver took it out then on return said it was the worse case he had ever seen outside the factory, he spent 30 mins or so adjusting the damper blades and shims then Perfect!
I had done this myself many a time to no avail. I think he may have checked the wheel balance etc.also but the damper blades appeared to be the main cause. It cost me £15 cash then, probably 2/3 days wages at the time !
It did return now and again after time but I was able to control it. 'Hope you get it sorted,do the above plus the usual checks, keep us informed!