I'm allergic to all face wash?

Every time I use any brand of face wash or moisturizer, my face gets EXTREMELY dry afterwards. It's so dry that it almost had the texture of rubber. I wash it all off afterwards so it's not that I'm not washing it off good enough. Any ideas?
Update: Another thing to add is that I use an eye cream to try and get rid of the bags under my eyes, but this also just makes it worse, but instead of getting more dry it gets more soft.
Update 2: Sorry for all the updates, but I don't know if this could be related but I also seem to be allergic to chapstick and it just makes my lips more chapped.
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