Hi. For about the past month, I have experienced pain
Hello, I'm Dr. Branch, thanks for using Ask a Doctor. This is not a symptom I have heard a patient describe before, but every patient is different and you might be surprised how often that happens. First, this doesn't sound like a symptoms of any disease that would be serious, so I wouldn't worry about that. Saying exactly what is causing this would be difficult, but one possibility would be due to something called referred pain. The different nerves in our body detect certain stimuli, and then transmit that message to the spinal cord, and then to the brain, and the brain interprets the location of that stimuli based on what nerves bring the message. However, sometimes those nerves can travel close to other nerves, and the message gets confused, and the brain thinks the pain or sensation is coming from a different part of the body.
I would suspect this could be the case for you. When you drink something cold, the nerves in the esophagus and stomach detect it, and somewhere along the way the message gets confused, and causes discomfort in your shoulder. In this case, moving and using the shoulder would not cause any pain, because there is not actually any problem with the shoulder, it is the nerves in the esophagus and stomach, which are sensitive to cold for some reason.
I hope that makes sense somewhat, please let me know if you have any questions about any of that and I would be glad to explain anything further.