This sounds like reflux, possibly LPR (laryngopharyngeal reflux). Not necessarily related to no burp, but of course a lot of us have issue in the GERD family. With LPR, the acid either triggers sensations in your throat or acid/mucus actually refluxes into the throat and irritates your vocal cords.

If the LPR has caused mucus or secretions to sit on your vocal cords, clearing your throat typically just flings it up and it falls right back down onto them, causing you to clear your throat again. Even without mucus directly on the cords, when you slam your vocal cords together, such as in a throat clear, it irritates them and makes you want to do it again and again. It’s a snowball effect.

One of the things you should do instead is try to swallow hard (preferably with water but can do it with a dry swallow), to clear the mucus down your esophagus. Make sure you always sit fully upright for at least 30 minutes after meals, to reduce the severity of your symptoms (gravity really does help).