Unfortunately, dairy and soy are in tons of packaged foods, drinks, and vitamins/supplements/medications. You really need to read labels on everything you put in your mouth. I'm dairy, soy, and nut free because DD2 is MSPI and has a nut allergy. She developed eczema after I was already dairy and soy free. I cut out all nuts, and her eczema was gone a week later. (Testing confirmed she's allergic to cashews. I was eating walnuts every day, not cashews. So I cut all nuts and that did the trick).
Anyway, DD2 was even sensitive to soybean oil and soy lecithin, which are supposedly okay for most MSPI/soy allergic people. So I'd avoid soybean oil and soy lecithin for a good month, and once LO's symptoms clear up, you can trial those ingredients and see how he does. Most peanut butter contains soybean oil.