By tranny axle do you mean driveshaft or a shaft inside the tranny?

Does it still vibrate for a bit after the car stops, or only while in
forward motion?

Have you been under it yet to inspect mounts for looseness?
Grab the driveshaft either side of the center mount and shake it hard.
If you can get it to clank, you have a rubber mount headed south.
Inspect the rear tranny mount while you are there. They get soft and
squishy if there is oil on them.

Make sure the driveshaft is aligned so that the yokes on each section
line up perfectly. Here is how it should be:


If this is not right, you will get pulsation while rolling at low speeds.
--
George Downs Bartlesville, Heart of the USA!