The "Hula" Mast...

[Brion Toss]

Hello,
I will guess that the prevailing wind blew from a different direction in your old marina. When you say that it was "off the stern" in this storm, do you mean dead astern? A broad reach? Nearly a reach? The kind of shaking you are describing is usually associated with vortices forming as the wind passes the mast, particularly side-on. When the formation and collapse of these vortices is harmonic with the mast's own moments, you get the shudders. Very hard on things in the long run.
Tuning will probably have a negligible effect on the problem, but do try hoisting a boat fender, with downhaul, on the main halyard. Take it up slowly, and the pumping will very likely stop when it reaches somewhere around mid-mast. Really. If this works, we can suggest other, less odd-looking procedures.
Fair leads,
Brion Toss