How crazy you need to be will depend on a couple factors. First, whether you have celiac or a gluten intolerance. If you have not been tested for celiac disease you will want to get that done ASAP as eating small traces of gluten as a CD patient can cause major problems with co-morbidities down to road.
Assuming you have been tested for CD and those test have come back negative the strictness of your diet is completely up to you. With the very limited research and knowledge of gluten sensitivity everything points to is as being an acute reaction. There for the only “price you pay” for eating gluten is the symptoms you get (sounds like in your case asthma reaction) when you eat gluten. So basically you need to be as strict as you want to be. You do not necessarily have to completely eliminate gluten from your diet as CD patients do, it is trial and error to find the amount of gluten your body can handle without cause symptoms. In some patients this is trace amounts (like celiac) and in others it can be much more.