I like Pre de Provence and L'Occitane Cade for soaps with decent moisturizing. MWF is good, but as I mentioned on another recent thread consistent usage gives me break outs so I never use it back to back.
Do a search and you'll find a ton of different opinions. Sensitive skin is very much a YMMV kind of thing.
First, make sure your lather is correct. Not enough product or too much water and you end up with light airy lather that allows your blade to take off more skin than necessary. Too much product and not enough water and you have concentrated soap on your face which will dry anyone out. If it's neither of those then you may be sensitive to the fragrance in the soap.
In general, I agree with giggle. Try a facial moisturizer and see if that helps.
FWIW, I've had very good luck with Pre de Provence. I've tried a bunch of soaps and creams and it's still one of my favorites.
Nivea Sensitive Skin Balm is great. Drinking plenty of water will also help with dry skin. I have fairly sensitive and dry skin, but most shaving soaps work well for me (except Cade), and creams tend to dry me out for some reason.
Queen Charlotte Soaps has a very nice puck that works for me. I ordered the sampler; Rose, Vetiver, Provence, QC, and Bacilica. None of them have caused me any irrataion, and leave my dry skin feeling buttery after I shave. Sensitivity is definitly a YMMV thing, for example Cella, AOS(even unscented), and some coconut oil software gives my skin fits to varying degrees. A slight duck sideways, the Speick stick works well for me also.
Vitos and Aos i find them both very nice and for moisturizing the new mystic water shaving soap has to be the most moisturizing soap ive used thus far. I really like Gft cream alot but didnt mention it at the beginning because its not a soap.