I had a broken coupler on my Whirlpool as well, so I replaced it yesterday. During the spin cycle, I too smelled a burning smell.
Does this happen because the two white pieces (with three prongs) have been driven too far onto the shafts of the motor and transmission sides? Is that what you guys mean by the coupling binding too far up against the transmission?
The plastic couplers look to be a bit improved from the replacement one I took off, because it has a metal ring in the center now. I will try JWWebster's idea to use the old black rubber ring (which is still intact). And how were you using the nut driver handle to get it on just right?
I was all prepared to go buy some 80w90 gear oil, because I also lost some oil before I fixed this because I had the washer upside down on its tub for 2 days to get easier access to it. I guess the oil slowly leaks out of the gear box. I thought that was the reason for the smell, but maybe it's the coupler. I will try repositioning the coupler first.