I suppose it's possible laundry detergent could be a trigger. For me it did not turn out to be. Both Tide and All make "free & clear" versions for sensitive skin folks. I have used both for over a year and still use the All. Unfortunately laundry soap was not the "silver bullet" solution for me. Using Cheer and a double rinse would seem OK, but it's extra time and money. Personally I don't think the caveman approach with water only / no detergent would work too well (i.e. clothes not very clean).
It was a huge factor for me. I found out I am allergic to sulfates (the stuff that makes soap sudd) and got bad breakouts all over. I now use all natural soap made from orange peels. I have fewer breakouts and less painful bumps. Some people really are sensitive to it, just remember that detergent takes up to 10 washes with clear water to come completely out of clothes and your washer and dryer could take longer to clear of it. I've also cut out wheat and gluten from having such a big role in my diet. I could never give up french bread or pasta. Keep track of your salt intake too and make sure you aren't using too much, if any, spray deoterants or "axe" type sprays since they can cause breakouts when you sweat. I'm allergic to those types of products myself and most people I know are sensitive to them, so they are used sparingly or not at all. Be careful how much fabric softener you're using as well as dryer sheets. Those also have chemicals in them that can cause breakouts.