I can't say that I can armchair diagnose your problem, but I can shed some light on the subject.
First, the clunking you have heard about from the rear ends of C3's is when the positration plates are slipping and grabbing inside the diferential, but this only occurs when the diferential is HOT, after driving a hundred or so miles. Even then it only occurrs when you are going slow, and turning. Not when putting into gear, as in your situation. That diferential noise / clunking can be very loud, and actually make the car jolt a little, kind of like a full time four wheel drive when you are turning. The clunking is caused by a break down in the viscosity of the rear end grease, 90 wt I think. I was told (whay back then in 1978) that the origional gear oil was made with whale oil, and when the US stopped harvesting whales, an additive had to be made by GM because the oil was no longer available.
In your situation, I believe what you are describing is called "drive train wrap up." The primary culpret is usually simply that the idle speed is to high. If the choke/ high idle is still on when you are putting it in gear, then try and tap the idle down with the gas pedal, before you put the car into reverse or drive.
If that is not the problem, then check the actual idle speed itself, the rpm info should written be on the bottom of the hood, or the core support. You can test this theory by actually lowering the idle speed lower than factory settings, but just as a test. You may have to overhaul the carb to correct the problem.
However, it is also likely you have a combination of things causing the problem. There are 6 universal joints in the drive line, and there are four caps on each u-joint with maybe 30 or 40 needle bearings inside each. Any one needle bearing can cause the clunk, or more likely the overall wear of all of them is cause. The clunk will continue, and get worse, including when driving, if the u-joints are bad. They are not hard to change with a floor jack and a bench vice. But be carefull in installing the half shafts, the u-joints have to be carefully be "pulled" into place by the securing screws at the spindles. Also, you will need a star socket for the bolts, unless they have already been changed.
Hope this helps.
