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This does indeed sound like a strain, swallowing if moving your chest wall muscles can indirectly move some of the back, therefore this could explain this sensation on swallowing.
You could try anti-inflammatory drugs such as Ibuprofen to help with this, but should continue to resolve over the next week or so.
An ulcer would cause pain at the front, most likely in the epigastric region, not where you suggest.
Muscle injuries can indeed take a good few weeks to settle
Certainly not, I would perhaps speak with your GP given your anxiety about this.
Given the timing and history, yes, this is what it seems. A muscle strain can take this long and sometimes a bit longer.
Yes, I do believe you should