Transmission compatibility
Duke -
The transmissions are definitely compatible between the two Lady Ks, however if this were my job to do I would be aware that the transmission in the '72 may not be in complete re-usable shape, AND as stated by others above, is not an entirely simple procedure for a novice.
My first recommendation is to get a Kenmore or Whirlpool produced repair manual, which will outline the removal and re-installation procedure. You should have one of those anyway if you're going to rely on a belt-drive washer. Then, remove the agitator in the '72, or at least try. Do the same, or at least try, in the '66 as well.
Both the '72 and the '66 are, at least in well used machines, prone to having rusted-on agitators, which will be very difficult to remove. The 72's can be taken out with the aid of an Agitamer, however if stuck, this will often reveal very rusted agitator splines, meaning the shaft cannot be re-used, and thus rendering the transmission useless unless you have a new agitator shaft. The '66's agitator cannot be removed with an agitamer, not in my view anyway, and if REALLY stuck, can only be removed by breaking it out with a large screw driver or similar. You'd have to use another agitator other than a Roto-Flex at that point unless you have access to a spare.
If you're lucky enough to have one or two agitators that will come off their splines by pulling, you're doing well. The swap job after that isn't that bad, however as Mark was eluding to, transmissions that have worn themselves into place need considerable coaxing to get out, as spin tubes can wear grooves onto the agitator shaft, and the centerpost can wear grooves into the spin tube, thereby clamping everything together in ways they weren't when assembled at the factory.
If you get to the point that the '72's transmission is going to go into the '66, also as mentioned above, the quiet-pak drive pulley on that transmission must be removed in favor of the '66's or similar pulley. All four pulleys (Motor, pump, basket drive, and transmission) must all be for the same belt, either way.
Start with trying to get out the agitators and see what you have at that point, then we can re-group.
Gordon