Will oxiclean do a good job of making my old dingy white shirts look white again ?

If you use it properly it will help a lot! You must let the shirts soak at least 8-12 hours in the proper concentration of "all color bleach". Oxyclean is just one brand of the stuff. There is also Biz, Chlorox2, Vanish, Blanco Colors and several others depending where you are at. Put the shirts in a plastic bucket of water to do the soak. You can also do it in a top loading washing machine but run a rinse cycle empty before doing the soaking. You must also mix the clothes around every 2-3 hours or so to make sure they remain in the bath evenly.