I've never considered myself somebody with sensitive stomach and the same can be said about my DH but we get diarrhea from the water in FL. We brought these neat refillable water bottles that had loops on them so we could hook it on our belts or my purse. We brought the CL packets. Within a day we had tummy troubles. So we stopped the water and stuck with pop. Problem cleared right up. We figured it was a fluke and probably not the water. So we brought our bottles again and filled up in the parks. Here came the tummy troubles. FL was having a serious drought at the point so we blamed it on that.
Next trip, we tried the water again. And here came tummy troubles. When I brought my SIL for her first visit, I had told her ahead of time about the water. I didn't pay attention to what she was doing but she told me that she had tummy troubles. I said "Are you drinking the water from the fountain". Yep, it effected her as well.
When I travel with my DH we rent a car. We stop at a grocery store and pick up bottled water. When I travel without renting a car, I have NetGrocer deliver bottled water to the resort. Much cheaper than Disney price.