Interesting.. While in Neutral, let the clutch out, the whole Trans and engine start shaking, or JUST the shifter stick???
Remember the 300/6 is an externally balanced engine.. Requires a specific flywheel, balanced for the engine, and Pressure plate as well.. Although many Fords have the same Crank Flange pattern, so the wrong Flywheel will bolt in place easily.. And that will cause this problem.. But it should be shaking ALL the time if that was the cause..
If it is just the stick, then it might be bent shaft in the transmission..
Hmmmm???