Both my boys seem to have developed very sensitive skin, DS1 being the worst of the two.
Currently I am using Dr. Bronner's baby bar soap every other day on them in the bath (although w/ summer and constant dirtiness as well as sunblock we will probably be doing nightly baths). I switched from using any kind of lotion (everything made him scream) to using only coconut oil on his skin (seemed to help for a while, now we are back to issues). They both are only sleeping in cottong at night since I noticed their fleece jammies were making them irritated more. I don't let them overheat. We use a liquid form of homemade laundry soap (dr bronners, washing soda, borax & vinegar as a softener). I'm at a loss as to what is causing the issues now.
The one thing I haven't tried yet is cutting out foods from his diet. I've had a number of people say to try cutting out wheat or dairy at this point (we are carb junkies and he loves cheese so this is going to be difficult if we try this). But his pedi also told me that allergy info just got tossed out the window since most kids don't accurately respond to allergy tests until age 3-4.