Feeling something stuck in the throat is called globus. GERD is one of the causes. Since you burp also, GERD is the likely cause.
Take PPI for relief.
A 24 hours pH monitoring test may be done for confirmation.
If a doctor cannot see anything in your throat such as swollen tonsils or other structures, you should have an endoscope. An endoscope is a "looking scope" that goes down the esophagus to see what's going on there and in the upper stomach. An endoscope does not hurt and has no preparation.
Burping likely would have nothing to do with tonsils unless you are swallowing excessively. Burping can be caused by several things. But you need to have your throat looked at to make sure there's nothing going on there--tonsil or thyroid problem. If nothing is wrong there, the burping and stuck feeling suggests GERD and/or hiatal hernia. But the endoscope and other tests will tell you. It's unwise to guess at these things without following up with tests.
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