My daughter & I have sensitive skin. We moisturize daily with Aveeno products or organic coconut oil. My baby bathes 2-3 times a week so we don't dry out her skin. I wipe/ clean her face, hands & feet daily before applying lotion. I wash her clothes and mine with Dreft laundry soap, also we don't use dryer sheets, we have dryer balls to help reduce static.
We did run the wash through with vinegar first then ran it through with detergent after sometimes run an extra rinse cycle. We found that if we get new clothes we do that.