Help? (details inside)?

Im 15 and got over a cold/sinus infection about 2 and a half weeks ago. I still have the cough from it and loosening up from my throat. Like ill cough up snot from the sinus drainage I had. I always get this. But today it was extremely cold and I was out walking to lose weight and I notice that im coughing from the "dry air" I think? lol. I dont have ashtma or anything wrong with me. I think its from the dry cooler air? Can it cause that?