I forgot to mention in my previous reply, that, when it comes to making a car secure, spare no time or expense whatsoever. You don't have to go out and buy anything elaborate or expensive to block the car up, BUT whatever you use, make damn good and sure that your blocking mechanisms are going to make the car defy gravity until the job is finished. It's best (good, anyway) to have the car raised from under all four tires, by equal sized blocks. LARGER railroad ties cut into sections make good blocks (if the creosote has kept the tie solid over the years) if you can find them. Whatever you use just make sure that the blocks themselves and whatever means of ramp you use to drive the car onto said blocks are solid and capable of taking the weight. You could also jack the car up and put the blocks under the tires (or solid supports underneath car) individually. Whatever means you decide of blocking the car just make sure it's safe and secure, and you might want to have someone guide you if you decide to drive the car up. As far as tools go a transmission jack is always handy-maybe you could borrow or rent one. Good luck and stay safe.

Which tranny: AW or ZF?

Check fluid level and condition: it should be red and not brown. Adjust the kickdown cable so that the shift point is correct. If these don't correct the condition, you are indeed likely in the market for a "new" used tranny. Rebuilding the old one is not wise when the market for nice used units is only $350 or so. The job is best done while the car is on jackstands, not ramps, for better clearance. You'll need a tranny jack. If you do it, read the 700/900 FAQ for details and plan to replace the rear engine and front tranny seals at the same time.

There are two possible answers to your problem, although be assured not the only possible two. There is a well-backed-up theory that, if the clutches in an automatic transmission are relying on "built-up" varnish as their last means of providing traction, the loss of said varnish can cause the clutches to fail. If your transmission is high-mileage, it's possible that the previous owner flushed out the old and brought in the new - Fluid, of which has detergent properties strong enough to "strip-off" the varnish that, unfortunately, became the lifeblood of the trannys' clutches.
Number two possibility is that your downshift (I'm almost sure that's the name)cable has become kinked and needs to be freed up\and or adjusted. The latter of the two would certainly be your better friend BUT this usually applies when the tranny only shifts at high RPMs, and no slipping should be involved. There was something about the way you mentioned "back off the throttle and it's OK" that made me think of this as a possible culprit. Good Luck.