I think I know
exactly what this is - and it's not even expensive to fix. =)
The "clunk" is the giveaway. Your u-joints are shot. When you put it in drive, there's slop in one (or more) of your u-joints, so the 'clunk' is the play being taken up. If you gently accelerate, there's no noise, or not much, but there may be vibration because things are out of place and not balanced. When you let off on the gas, CLUNK again as the play in the drivetrain goes the other way - wheels driving engine. Then you step on the gas again, and CLUNK, now the engine's driving the wheels again. Classic. With the car up on jackstands, you can just grab the driveshaft by hand and wiggle it back and forth to see how much play the U-joints have. It should be "none".
As for the 'one wheel spinning' thing, this is the job of your differential! You might expect them to both go equally, but they have to be able to spin at different speeds so you can go around corners. When you turn, the inside wheel must go slower than the outside wheel, or else one will have to scuff and slip. The differential is kind of a neat thing. If the car's in park (driveshaft can't move) and both wheels are off the ground, you can turn one by hand, and the other will spin the opposite direction. If one is held, and the driveshaft turns, the other wheel will spin at double speed.
If you want to see more and understand the dark sorcery behind this nifty gearing trick, here's a
video:
Definitely fix this - you don't want your car flinging the driveshaft out on the road while the car's moving! It could cause a wreck, or major body/mechanical damage, or both! But it's not hard to do.
Welcome to AFM!
Edit: DOH - apparently I did miss a very important line here. I see that you did fix the u-joints! >.< PMAZ is right to check the slip yoke next. It should slide in and out easily just a little, but should have zero play in any other direction. If there's any slop at all, something's wrong with the tailshaft or your slip yoke.
The last thing it could be is your rear gears. Does it growl or whine out on the road? Usually you'll hear things going a long time before they're shot, but if there's a problem with the pinion gear, you could test it for play with the rear wheels, and by wiggling the diff end of the driveshaft.