Am so happy. The internet is the most awesome place in the world. I can learn Devanagari in one hour, and learn how to swallow a pill in two minutes.
Long story short, I found another blogger's story of pill-swallowing-inability (hat tip: Ennis), and about halfway down the comments there is a suggestion to try pressing the tongue completely up towards the top of the mouth, which kind of forces everything down at once.
The way you are supposed to swallow is that you close your jaw (back teeth touching) and (MOST IMPORTANT) push your tongue up against the top of your mouth. This makes it impossible to breath in air through your mouth, as it creates a vacuum seal and then the suction force easily pulls the food down in to your stomach.Anyway. For whatever it's worth, it worked for me. After how many other methods of trying.
Woo-hoo! Now I won't get malaria!