Suggest Remedy For Chronic Dry Cough

My son is 9 yrs old. Since Thursday I notice him coughing frequently and has a cold. Today I started giving him Mucinex for cough only. He does not have a runny nose but he blows his nose often. No phlegm coming out (that Is what my son says) when he is coughing but I can hear his cough with some sort of mucus. Am I giving the right OTC? Took his temp a couple of hours ago and reading was 99 should I give him Tylenol?