2001 Mitsubishi Eclipse transmission sounds

I just replaced my F4A51 transmission with another one and now when I put it in reverse or in gear it sounds almost like a dry bearing sqeaking. I thought initially it was a pulley so I've torn it down to the tiiming belt and checked all the pulleys, water pump and such. I've driven it a couple hundred miles and it shifts smooth and the sound is barely there and only at idle. The other thing I noticed is that when I turn the engine off it sounds like something rubbing for a couple of revolutions until the engine stops turning. It almost sounds like the fly wheel rubbing or something like that except it really only makes the sound when put in gear, it goes away in N or P. I thought when I first drained the tranny I found a piece of the wave spring but thought that it may be from my old tranny since I had it torn apart to see what the problem was and it had crapped the wave spring ruining the pump. Will this transmission run with a broken piece of wave spring? I'm not really good at diagnosing or even working on transmissions except to replace them. Any ideas or help would be appreciated.