Since you have sensitive skin, you may want to consider using
Aveeno lotion. When my nephew was in ICU for a week after he was born, they used Aveeno on him. I figured if they use it on very weak and mostly premature babies, it must be gentle!
I am allergic to pretty much everything you'd buy in the store (shampoo, conditioner, laundry soap, sunscreen, hair products, etc) so if you need any suggestions, I've used a lot of products in my trial-and-error routine!