Thanks for all the replies. Dove soaps, at least the ones they currently carry, are not real soaps made from oils, but rather solid forms of detergent body wash. Check out their lists of ingredients and you'll find them strikingly similar to their liquid body wash. I believe the same is true for Olay and J&J. Yes, they are not basic, and I don'y have allergies to them, but I'd like to try my luck with other soaps before I revert back to detergents
The best scenerio is that I'm allergic to some fragrance or additives in that soap, the worst would be that I'm allergic to all natural soaps. I've thought of side reactions too...but given that my facial moisturizer is thicker/greasier than my body lotion, it's also very likely that the soap is just too drying, and my lotion is not up to the job to keep the moisture.
Again, thanks for all the advices