Just an update. Since I would have had to buy a metal tailpiece, and a new connection for the dishwasher's drain hose I decided to give the plastic nut and tailpiece one last shot. I ended up using the plastic nut and tailpiece with no tape on the metal strainer. After a couple tries of tightening it down, it seems to be holding.
Glad it is working for you. Through this whole post, it seems to me that you may have had a crooked connection to the strainer, which was causing the leak. Most threaded PVC fittings do not need to be "cranked" really tight. This usually causes the nut to crack or strip and do exactly what yours did.
If you have never made a mistake, you haven't done much.