How many miles on the motor and how many since the last tune up?

I suspect one or more of the ignition components: coils and / or the
spark plug connectors, especially if the shake rattle & roll occurs cold
and hot at both idle and during power demands.

Any codes?

Our W124 chassis has a known problem with Overload Relay Protection... if
you never changed it before, I'd start off with this one. The erratic nature
of failing OVP can cause numerous symptoms including the one you are having.

Looked at the spark plugs.  One of them was caked in oil, no idea how
it got in there.  There doesn't seem to be any leaks so it must have
spilled in there somehow.  I cleaned it up the best I knew how, but the
plug needs to be replaced.  Any tips on cleaning the oil out of the
hole?

After cleaning the plug the car ran better, and after a few seconds it
was back to it's old self.  But once I turned it off and back on again,
it started back to shaking only now it won't shake in P or N.

I still suspect the overload relay may still be a problem.  Is there a
reliable way to test it with a multimeter?