Hello~It does appear to me that you have a sensitivity to wheat, which means usually, you will have other allergies/sensitivities as well. I am also wondering if you have "Candida" This is a type of yeast infection that starts in the gut but can go anywhere in your body, it has all kinds of symptoms. I suggest that you cut out all sugars, yeast and of course the wheat for a few weeks and see how you feel, do some research on Candida, you will be surprised at what it can do to some people.
Here is a link to some info about it, I hope it helps.
Having a plant-based diet doesn't mean you have a healthy diet. No matter what one chooses to do with one's diet, it has to have the proper amounts of nutrients to be healthful. If eliminating wheat helped that much, it is possible you have ciliac disease. That would require the elimination of all gluten-containing foods, which would include wheat but also some other grains. It could also just be a wheat allergy or intolerance, which is much more common. I'd eliminate dairy as well if you consume it just to see if that helps as well. Other common allergens are soy, corn, and for some people certain nuts and for others another legume, peanuts. I'm just mentioning this because cutting out wheat helped. Often people who work in nutritional or older forms of medicine will tell everyone with any problem to cut out wheat and dairy and see if that helps fix the problem. That being said, this doesn't exactly sound like those problems, but for you maybe it is. Given you say you eat a plant-based diet, how are your B12 levels? There is almost none in plant food. Another essential nutrient hard to get in plant only diets is methionine, an amino acid. These two nutrient deficiencies probably explains why there is only one vegetarian civilization we know of, which are Hindus, but they eat a lot of dairy so they obviously knew about this. The only vegan people are the Jains, a small religious group in India. Both groups are known for short lives, as are the lives of those who eat a largely animal based diet. It seems the best balance is moderate animal favoring animal food from the ocean and as much of a diversity of plant food as one can get so one gets a wide variety of antioxidants. I only mention this because of the possibility of a B12 deficiency. Barring that, I have nothing else to offer but to say, if your current docs can't help you and it's not dietary, get better docs. Peace.