Hi there. Much sympathy, this sounds stressful. A few possibilities strike me. One is a cmp allergy. Another is reflux, though this is often a symptom of a dairy allergy. And another is that it is two separate problems - sensitive skin as well as some sort of digestive problem. Another is another food allergy. Most common I think are dairy, soya, egg, gluten, legumes (peanuts, lentils, peas).
In my experience of two allergic kids, food allergies go hand in hand with excema and dry skin. The skin takes a lot longer to clear up than the tummy trouble. I'd give it at least a week from cutting something out from your diet to see any difference in your baby if it is an allergy so be patient!
Maybe try a very bland allergen-free diet for a couple of weeks and see if that helps, then reintroduce foods one at a time? There are lots of threads on here with meal ideas for various allergies. As a start, stick to potatoes, meat and root veg for a few days until you get your head round it all!
And oat milk is a good replacement for cows milk - soya can make you v windy in my experience,as well as being a common allergen.
Surcare is a good laundry liquid for sensitive skin and others suggest doing an extra rinse, something I have never got round to trying.
You mention trying a dairy free formula - if you can resist this for now and continue bf I think that might be better, just because you can totally control what your baby gets in his feeds that way, and the only df milk you don't need prescription for (and prob won't get a prescription for without dietician referral etc) is soya formula and that's not suitable for babies under six months. Also, even prescription milk doesn't agree with some babies so if you keep up the bf you know he will always be able to have some milk!
Good luck x