What can I do to get rid of chest and back pain when I eat and swallow?

GI issues. Chest and back pain related to eating are most likely related to problem with your GI system. Although treating acid reflux usually takes care of these symptoms, it is highly recommended to follow up with your primary care physician in order to rule out more serious problems (e.g. Heart disease, pancreatitis, gall stones) and to suggest a plan of action including referral to a GI specialist.

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What should I do to get rid of chest and back pain when I eat and swallow?

Get eval. The cause necessary inorder to treat. If you can find a way to go to a GI doctor, I think you would find your answer.