Also sensitive. It's just my skin rather than breathing in the scents.
Have never yet found the fragrance-free products I need in Italy. Even baby soaps and products labeled hypo-allergenic have scent in them. I have tried looking in giant hyper-markets, stores in big cities, several pharmacies, and natural foods stores. Googled quite a bit, too, only to find other people have the same issue.
I do not usually check my bag, not even for a three-week trip. I carefully figure out how much powdered laundry detergent I need (Italian washing machines are small and efficient and take only about a tablespoon or two of the powder, which I mix with hot water in a glass to turn it into the required liquid). I also figure out exactly how much scent-free bar soap and bar shampoo I will need. Likewise Aveeno hand lotion although I have seen that for sale in pharmacies. I take a baby sunscreen that's in a stick. I don't use conditioner, and I must be less sensitive than you because I don't worry about dish detergent as long as I wear rubber gloves (available in Italy). So, only the Aveeno and various skin medications need to go in the 3-1-1 bag.
Twice on Italy trips, the sheets and towels where we stayed have reeked so badly of perfume that we had to wash them before we could use them. You might want to check with your accommodations.
Something you might want know: Italian washing machines can take hours to do a small load of laundry and some cannot do an extra rinse.
I've never dared to try it, but I've read that many clothes get pretty darn clean in a washing machine without using any detergent at all.
I wonder if Amazon would deliver some of your products in Italy.
Get a window seat on the airplane(s).
Remember that you'll need to take your hand soap with you into all the bathrooms you encounter.
Good luck --- Italy is absolutely worth it!