Hi..
Welcome to HEALTHCARE MAGIC..
I have gone through your query and can understand your concern..
As per your complain coughing after exposure to dry air can be due to irritation in throat caused by inflammation and it can lead to
sore throat and irritation in throat can cause
tingling sensation that can initiate cough..
Due to vigorous coughing there can be trauma to mucosa of throat and can lead to slight bleeding..
I would suggest you to do warm saline gargles.
Take anti allergics like Levocetrizine..
You can also take cough suppressants containing
Dextromethorphan and Gueifenesin..
You should do steam inhalations..
You can also sip warm water.
Avoid spicy foods to prevent further irritation to throat..
In case if there is no improvement in a week or symptoms worsen consult an Otolaryngologist and get evaluated..
Hope this information helps..
Thanks and regards.
Dr.Honey Nandwani Arora.