Since you are posting this question on a thyroid disorder discussion board, I assume you think it might be a thyroid problem.
If they've done chest x-rays I would assume they've also felt your thyroid and possibly ordered an ultrasound (since my doctors and dentist are constantly feeling my thyroid as part of the standard patient exam before they remember I don't have a thyroid).
An enlarged thyroid or a thyroid nodule can put pressure on both your esophagus and trachea, which can make it hard to swallow food (and may affect your breathing). If they have not done so, I would suggest you request a thyroid/neck ultrasound and a test of your thyroid hormones (TSH, freeT3, freeT4). Most thyroid nodules are benign and may not need treatment, but if it is obstructing your ability to swallow or breathe, it may need to be removed regardless of whether it is benign or malignant (if the obstruction is caused by a nodule or goiter, that is). I have Hashimoto's (an autoimmune disease where immune cells attack(ed) my thyroid), and my neck was full of inflammation (lots of swollen lymph nodes in addition to a multinodular goiter) when my thyroid was removed. It is possible there could be inflammation in your neck pressing on your esophagus, which they might be able to tell by thyroid ultrasound or palpating your neck.
There are several structures in the neck that the doctors should work to rule out... inflammation of the esophagus itself can cause problems, an ulcer of the esophagus can cause similar symptoms to what you have described. Have they explored your esophagus (or trachea?) with a camera down your throat/nose? I know they can inspect larynx and vocal cords this way, and could possibly rule out esophageal inflammation/ulcer. Acid reducers might help with esophageal inflammation (this is well outside my knowledge base) but if there is an ulcer in the esophagus I doubt acid reducers would fix that. Vocal cords also can get polyps, which I assume could also impede the esophagus, much like thyroid nodules.
I have no idea if it actually is a physical obstruction in your neck, and we would not know until after they do an ultrasound, but it sounds like this is greatly impeding your ability to eat and they should at least rule out a thyroid problem by ultrasound. If they've already done that, it is possible it might be a "lump in the throat" sensation that is not caused by a physical obstruction in the throat.
Other potential causes of a lump in the throat sensation:
Loss of muscle coordination
Post nasal drip
I know this was just a list of possibilities (and its possible the cause isn't listed here), there is no way to know what the cause is until potential causes are ruled in or out.
Have you been to an ENT? My ENT (who removed my thyroid) did a test to make sure my larynx was undamaged, before and after the surgeries (with a camera in through the nose). If there is something physically pressing on the esophagus when you swallow, they should be able to see this with a camera down the esophagus. I have a neighbor who has had esophageal biopsies on polyps, which could also cause similar symptoms.
I'm not sure any of this is helpful, but good luck, and I'm sorry it is taking so long to figure out what is going on. That lump when you swallow feeling sounds very uncomfortable.