It is more likely that they meant to cut out gluten (the protein in wheat products) than wheat, per se. However, before you decide to do so, I would examine what symptoms you are experiencing and cross reference those to common food allergies. Often, reactions to dairy and mild reactions to gluten produce similar symptoms -- such as sinus congestion and headaches.
If you are experiencing more severe symptoms, such as lung congestion, nausea/vomiting, difficulty breathing, you may be experiencing a wheat allergy, which is more severe than a simple intolerance. A wheat allergy can manifest in adulthood, even if you were wheat tolerant as a child.
Depending on your symptoms, you may want to take the plunge into a gluten free lifestyle and there are plenty of products now that offer gluten-free alternatives to foods that traditionally contain gluten. I recommend giving the gluten free lifestyle a chance for at least 30 days and judging for yourself whether it improves whatever symptoms you are experiencing.
Since everyone's body is different, what works for one person may or may not work for another. Only your body can tell you what will make you feel better.