What you describe suggests that you may have acquired a viral
pharyngitis, and this is often characterized by persistent cough, lasting for a few months
or more. Pharyngitis does not cause tightness of the chest but you may have developed some bronchospasm, in response to viral bronchitis. This could be aggravated, but not caused by cold weather.
If you were intubated during the sinus operation, you may have had irritation from the tube, rather than infection, as a cause of the pharyngitis.