Could being a caretaker for another person cause constant diarrhea?

Possibly: Irritable bowel syndrome can be worsened by stress. If the person you were caring for had an infectious diarrhea at some point, you could have caught it from them. If they are older and/or have been hospitalized a clostridium difficile infection would be a definite risk.

If they carry...: ...A diarrhea-causing infection like giardia or salmonella, then it's a possibility.