I found this site by searching "thirst" and "celiac," because I too am trying to find out about the strange thirst I have. I'm not officially diagnosed with celiac, but I have the same gastrointestinal symptoms and respond well to a diet that avoids the same things. (Actually, I avoid most carbs -- following the "special carhohydrate diet"). I've mentioned my thirst to a number of doctors and have gotten no explanations. I'm not diabetic. It's very odd.
(I do urinate a lot, but that's been going on for decades, whereas this thirst and gastrointestinal problem began within the past two years.)