Remember I am the one with true swallowing problems and speech due to the vocal cord paralysis. I have learned quite a few tricks which can help you with the "hairball"
stuck pill sensation.
I learned these from speech therapy.
Lower your chin and tuck it in when you swallow and using a straw helps.
Drink tons of water and knock off all the caffeine. It dehydrates you, and doesn't do all the throat irritation any favors.
I do simple tongue exercises which have helped tremendously.
There are many exercises for swallowing issues and one or 2 sessions with a speech therapist for "Dysphagia" may be quite helpful. If have a persistant problem,I am sure your surgeon or primary care doc will give you a RX for a speech therapist.