What you are dealing with is a condition called angular cheilitis, which is cracking and fissuring at the corners of the mouth. Keeping the areas moisturized, as you are trying to do with chapsticks is important, but often this is not sufficient to clear it up in chronic cases.
Many cases of angular cheilitis are actually caused by colonization of the skin around the mouth by yeast organism ("thrush"). Because of this, a topical antifungal cream is often quite effective at clearing it up. You could even try this as an experiment using an over the counter cream from your local pharmacy or consult with your primary care doctor
Rarely, cases of angular cheilitis can be caused by other bacterial species and can require antibiotic treatment. Similarly, sometimes angular cheilitis is caused by deficiencies in certain nutrients, such as B vitamins. Also, sometimes cold sores caused by the herpes virus can be mistaken for angular cheilitis.
I would suggest continuing with your regular lip care moisturizing routine and consider trying an antifungal ointment. If this does not work you will definitely want to see your primary care doctor to make sure nothing else is going on or another form of treatment is not needed.