I actually react far worse to dairy than to wheat. Before I gave up eating wheat and dairy, I also had really bad fibromyalgia. I first noticed the intolerance to wheat a few years ago when I took Flagyl (a very harsh antibiotic) because the doctor suspected I had bacterial overgrowth related to irritable bowel syndrome. I quickly developed an intolerance to wheat. When I'd eat cream of wheat I could barely hold it down, it felt like I was eating glass.
I do not have a celiac diagnosis. I got a colonoscopy and blood testing that came back negative. But I do have the HLADQ2 gene haplotype (I found this out courtesy of 23andme), as do a lot of northern Europeans, and the overwhelming majority of people with Celiac disease. The doctors have no other explanation other than I have a food intolerance to wheat and dairy.
There are wheat-free breads out there, though they cost 3-4 times a regular loaf of bread. My favorite is Mrs. Katz's challah bread, it's the best wheat-free bread I've found.
I eat a lot of traditional Hispanic and Asian food. I live in a predominantly Hispanic area and its not a problem to get lots of wheat-free food like tostones (pressed, fried cassava), carnitas (fried pork and corn tortillas), yellow rice, or tamales (corn and meat). On Sundays after church we sometimes go to the local Vietnamese restaurant downtown- Vietnamese food is all rice, vegetables, and meat of various kinds (and fish sauce). The rest of the week I eat at home. I buy some Luna snack bars and Chex makes some gluten-free oatmeal.
I have experimented a little with eating ancient species of wheat like Einkorn or Spelt and it doesn't seem to bother me in small amounts, but its still not something I feel like eating every day. The only kind of dairy I can tolerate is goat or sheep, cow milk has some kind of protein that makes my stomach upset, so mostly I use soy or hemp milk.