Anything with wheat, barley, rye or oats will contain gluten. The gluten in oats is slightly different from the others and doesn't usually cause problems although those who are very intolerant can sometimes be affected from cross-contamination because oats are often processed through the same machinery as wheat et al. Gluten is also present in beer, as thickeners, sweeteners (some dextrose is derived from wheat), hydrolysed vegetable protein, MSG - it can masquerade in many guises. For those who are extremely sensitive (just one crumb can put them out for weeks) it can be a bit of a nightmare until they have established which foods are 'safe'.
If you are not too sensitive then you might 'get away' with some carbs other than bread. I do wonder with bread, particularly certain brands, if they add even more gluten than has already been expansively (and indigestibly) bred into the wheat so that they can get more product per pound out of it, and that may be why you react to it.
Also, apart from perhaps stoneground baked wholemeal bread, the white stuff, particularly the 'pap' bread, often has extra yeast added to it to make it rise quicker, and flour 'improvers' which can include data esters made from petrochemicals. Whereas traditonally baked bread usually takes several hours from start to finish, the pap stuff can be in one end and out the other within 45 minutes! That ain't natural!
The thing is that if you do have a sensitivity to gluten, then even consumed at low level it can still be causing damage in your digestive system without you realising. I may be over-egging it in your case, but as you are still consuming gluten at the moment (you have to either be, or have been within the last 3 months consuming gluten for it to work), albeit a fairly low level, I wonder if it might be worth you asking for a Coeliac panel test. Even if it comes back 'negative' (it is not considered as possible Coeliac unless it is over 10), any positive reading, even a low one will at least show that there is definitely gluten intolerance going on.
I had to educate my Doctor about Coeliac and she has since done research and now knows a lot more about it. Most are still stuck in the dark ages with their understanding and think it only affects children who have a 'failure to thrive'. In reality it can manifest at any age and can be responsible for both under, and overweight along with a host of other health issues. Often by the time people start getting symptoms, damage is already established and malabsorption issues usually are present.
My Doctor said that she often has to repeat the test several times before getting positive results with some people - the testing process throws up an awful lot of false negatives. The reason they set the 'positive' at anything greater than 10 is because so many of the 'healthy' controls were showing a positive reading. They didn't seem to twig that not everyone displays symptoms of gluten intolerance, and that even seemingly 'normal' things like bloating and flatulence can indicate a problem. If they set the positive at anything above 0 then at least a third of the population would have Coeliac Disease and that would frighten the life out of them!
If diagnosed Coeliac you can get gluten-free foods on prescription, but to be quite honest, as most of them are very carb-dense there is not much point to that.
Of course, the best thing you can do by far is to stay away from processed carbs. The low-carb diet is good for that as the carbs you do get tend to be from natural foods, fruit and veg, maybe a little rice, pulses, etc., if your body can cope with them ok.
I have had digestive issues most of my adult life - I would get so weak sometimes that it would take all my strength not to just lay down in the street! Thinking back, it was usually after eating carbs of some kind, although I didn't twig the connection at the time. I used to get backache that I thought was because of my weight and now know was my liver and/or pancreas reacting. I would get acid reflux, hernias, bloating, gas, indigestion, you name it. Eventually I had IBS, mainly D, and the culmination was the awful, awful stomach pains every time I ate which was followed by the collapse of my digestion. In a way I was fortunate because I had symptoms and was able to figure it out (yes, me, myself and I - the Medical Profession were useless), but some can end up with 'silent' Coeliac like my Mum did - a host of (as I now know) linked symptoms, but no real digestive problems until a year or two before she died.
After dumping the gluten and most dairy all those, and other symptoms have either gone or have virtually gone.