Hi welshbec - welcome to GT!
My Cody always had a very sensitive digestive tract. She also had a few bouts of pancreatitis so NO fatty foods for her. I ended up finding a food she could eat & always added digestive enzymes (I used ProZyme, got it online MUCH cheaper than at vet's).
When her stomach would act up, the vet would immediately tell me to do the chicken & boiled rice thing. It didn't help & made her worse - I had one of the few dogs allergic to RICE!!! When I switched her to chicken & pasta, the big D got MUCH better. Then I had to find a food without rice... There's more of a choice there now than there was 12 years ago!
Good luck with this - once you find a 'cause', it should improve. You just have to find it...