Hello Heather,
do you still have your tonsils, or did you have them removed? The area you describe as your Adam's apple being swollen is most likely swollen lymph node and that happens when you have either tonsil or throat infection because lymph nodes sort of collect waste products your immune system creates when fighting an infection.
I don't see how a cough medicine could help, but try gargling your throat with salt water - it helps to numb down area a bit and it helps in fighting the infection, so it at least might be a bit easier for you to eat something. Also, maybe put the food in the blender first - it's not in any way permanent solution, but at least it might help until you get an enterologist to take a look and see what exactly is happening in your throat,
Wish you all the best,
Nicole