What is this gross taste in my mouth where my nose and throat meet?

Even though you do not have your tonsils anymore, I would not rule out a condition known as tonsil stones. Basically, they are buildup of food and other particles trapped and hardened within the folds in the back of your throat, generally where your tonsils would be. Grab a flashlight and a mirror and look at the back of your mouth. If you see white spots (not the kind you would see with strep, these are more like little white stones) then you could very well have tonsil stones. Sometimes you can get them out on your own by flexing those muscles or using a q-tip, but I would recommend seeing your doc before trying anything on your own. I hope this helps!