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Thank you for that detailed history and uploaded EGD report (attachment removed to protect patient identity). Reflux esophagitis does sometimes cause symptoms of inner ear pain, sore throat, and chronic cough, but it is much more common to have symptoms of choking during the night time and heartburn.
You have been diagnosed on endoscopy with a 2 cm sliding hiatus hernia (which is comparatively small size), grade A (LA classification) esophagitis. The polyp from fundus is usually benign (fundic gland polyp). A few biopsies were taken from esophagus, apparently to rule out eosinophilic esophagitis together with biopsies from duodenum and stomach. I guess you are awaiting the results.
To answer your questions:
1. Yes, Dexilant (Dexlansoprazole) mostly improves the symptom in two days. Dexlansoprazole will act immediately but relieving of symptoms may take some time because it depends on mucosal healing. Dexlansoprazole can cause minor side effects like nausea, vomiting, diarrhea, dizziness, etc. You should continue Gaviscon for a few more days. Avoid liquid diet, take more solids or semisolids, do not go immediately to bed after food, keep head end little elevated.
2. Esophagitis and reflux can cause breathing difficulty, but it happens usually if the symptoms have been persisting for a very long time and have been left unaddressed, so as of now I do not think you have any significant lung issues other than the probable occasional reflux of gastric content causing some symptoms.
To improve your current symptoms, my suggestion would be to add Gaviscon tablets after every major meal for a week, simultaneously continuing with your Dexilant, which I believe you are taking 30 minutes before breakfast.
The other lifestyle modification advice would be to restrict coffee and tea to 2 to 3 small sized cups for a couple of weeks, to not lie down at least for two hours after a major meal. Avoid wearing very tight fitting trousers or belts over the abdomen. Elevate the head end of the cot (or there are a special type of pillows which has a slant) by at least 6 to 8 inches. Avoid citrus fruits and sour food for a few weeks. Restrict smoking and alcohol.