I would suggest that you discuss your concerns with your Doctor or Naturopath. The symptoms you describe while seeming relatively simple may be indicative of other health concerns.
I have included some general advice on following a gluten free diet and the considerations to nutrients that need to be made.
Following a gluten free diet can predispose you to a number of Vitamin Deficiencies. The redness and dryness around the mouth may be indicative of a B Vitamin deficiency which includes but not exclusively Riboflavin. Coeliac disease and the effect it has on the villi and lining of the bowel will often cause nutrient deficiency. The recovery from these deficiencies and the absorption of nutrients from the diet requires that the digestive system be in a good state of health.
Naturopathy considers that there can be a range of predisposing factors that contribute to the occurrence and the production of the allergy or intolerance, these are particularly evident when Coeliac disease occurs later in life. These predisposing factors can include a history of Irritable bowel, stress and lifestyle factors, bad diet, exposure to chemicals and environmental pollutants, immune challenges and illness. The gluten in this case is considered to be a maintaining factor. A shift to a gluten free diet will remove the maintaining factor but does not necessarily address the underlying inflammation, and digestive repair required to lead you back to a state of good health.
The inclusion of a gluten free diet often sees a shift away from all grains regardless of whether they are gluten containing or not. Grains are an important source of B vitamins as well as essential fatty acids. Both of these nutrients are essential for optimal health and require a healthy digestive system for their absorption. In Coeliacs I suggest the use of a liquid B vitamin supplement. This ensures that absorption in maximised and provides a wide complement of B vitamins for the body. I also include the use of fish oil or a fatty acid supplement.
Redness and dryness are sometimes symptoms of a fatty acid deficiency. Good quality fats in our diets usually come from plant-based nuts, seeds and grains. People who have sensitive and reactive digestive systems will often shy away from their consumption to avoid digestive symptoms. Using a high strength fish oil like the Ethical Nutrients Liquid fish oil will provide high doses of essential fatty acids in a well absorbed form.
Zinc may also be important to aid with tissue healing and skin repair. I suggest the use of Zinc Fix, a high dose zinc and vitamin C powder which is easily absorbable.
It may be necessary to speak to your naturopath so they can assess your diet and any areas that may be lacking. The Coeliac Society also offers assistance to Coeliacs in regard to diets, recipes, support groups and food lists.