727 vibration

Need to run it up in neutral and see what it does, then drive it and carefully "feel" and note the difference. Could be converter/ engine balance, something in the drive belts, or out of true/ unbalanced drive shaft.

You could disconnect the drive belt(s) when engine is cool for a short time to eliminate them.

Also if the car is modified, could be a pinion angle problem

And of course don't forget tires/ wheels mount/ true/ balance problems