Your symptoms of right-sided nasal watering, sneezing, and headache after exposure to cold, dry air likely indicate an allergic or irritant reaction affecting your nasal passages. This is common with exposure to cold air triggering inflammation and mucus production on one side.

• These symptoms can be managed with anti-allergic medications like the combination of Montelukast and Fexofenadine, which reduce inflammation and block allergic responses.
• Avoiding exposure to cold, dry air and using a humidifier indoors may help reduce irritation.
• If symptoms worsen, persist beyond a few days, or you develop fever, severe headache, or facial pain, please consult an ear, nose, and throat (ENT) specialist for further evaluation.

If you want, I can explain how these medications work or suggest additional supportive measures.To narrow down potential causes, we will need some more information.Please tell me your age and gender.