Sounds like cough varient asthma. I would see a specialist.
Two possibilities come to mind: First, that you could have had pertussis, commonly called whooping cough. If so, the cough might take a couple months to completely resolve.
The feeling that your throat is "closing-up" is, indeed, a very scary sensation. This suggests the possibility of disease of your larynx, commonly called the voice box, or the epiglottis, the structure that normally closes off your trachea, commonly called the windpipe, when you swallow so that swallowed material doesn't enter your lungs or disease of your vocal cords, either anatomic or functional.
You should request consultation with an ENT specialist, capable of examining your throat directly with an instrument called a fiberoptic laryngoscope, to make sure that you don't have any disease of the larynx that could lead to progressively worsening obstruction of your airway.
The night-time awakening "gasping for air" is also consistent with the diagnosis of gastroesophageal reflux disease (GERD), which is regurgitation of abdominal contents into the esophagus with aspiration of the material into your larynx, causing spasm of the vocal cords. Either your primary doctor or the ENT specialist should test you for reflux.
Good luck.
Thanks so much! I'm going to DEFINITELY look into that.