From your description I'd suspect that the propshaft either has a worn joint or the centre bearing is worn or its out of balance
you can easily check for for any slack in the propshaft joints by twisting either half by hand and feeling for any play as there should be virtually none
the centre bearing is quite loosely mounted so this can be moved normally but if you can run the car in gear with the wheels off the ground and then feel the centre bearing housing for vibration if there's any then consider replacing the unit
and then look along the propshaft for any signs of a missing balance weight as these take the form of little 3/4" squares that are welded onto the shaft - as they are only tacked on they can sometimes come loose and be lost resulting in the shaft being out of balance - if a joint is worn this can also shake them loose
so if a joint is replaced its good practice to have the shaft re-balanced afterwards
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