IANAD, but based on my own experience, I think what you have is almost certainly Temporo-Mandibular Joint Dysfunction (TMJD). I had exactly the same symptoms as you describe, intermittently for several years. I went to a GP, an ENT specialist, and even had a dentist X-ray my wisdom teeth to see if it could be caused by an abscess. None of them found anything. Finally I read a magazine article about TMJD, and it mentioned that this was one of the problems it could cause. I went back to the dentist and told him that was what I had, and he agreed. He made me a dental appliance to wear at night, and that pretty much solved the problem. After several months I quit using the appliance, and have had only very infrequent reoccurences.
Until I pointed them to the diagnosis, none of the professionals I went to suggested that as a possible cause. That was about 40 years ago, so maybe they have gotten better in that time, but I suggest that you should be sure to mention it to them. If that is the problem, you would want to go to a dentist or oral surgeon to treat it. Since I am nearly positive that is what it is, my suggestion would be to go there first. Antibiotics will not do any good unless there is an infection, and overdosing with antibiotics is not a good idea.
As best as I can figure out, what happens is that when the side of your face is down, it pushes your jaw out of alignment (maybe even out of joint), and that causes pain in the joint, which is just under your ear, and you feel the pain in your ear canal. When it was hurting, my ear was very sensitive to touch. If I stuck my finger in the ear, it hurt even more, but after I was up for a few minutes, the pain went away. I expected to look at the ear and see that it was all red, but there was no difference in its appearance at all.