When transmissions go bad...
...they, grind, squeal, whine, hum, chatter, smell, balk, pop out of gear, refuse to go into/out of gear, leak, grenade, and yes, vibrate - but not so much to the degree you've described.
I've seen input shafts cause fairly severe vibrations, but those cases were related to clutch/flywheel/bellhousing issues that exhibited a host of additional symptoms.
You can't do much but speculate at this point - seems time to start getting up close and personal with the running gear.
Have a shop check it, or DIY by getting the rear wheels securely off the ground, sitting at the same angle as when loaded. Have an assistant run it thru the gears while you do the inspection underneath. The inherent dangers should be obvious, but if you don't feel safe/comfortable doing it, take it to a shop.