Maybe don’t over-do it trying to “stop” it - that stuff drips for a reason? Like, where else is it going to go?
I would actually suggest going the other direction and like drink a bunch of hot drinks to try to let more come out, especially at a time of your choosing (first: inhale the steam, then: drink, repeat as desired). I would guess that once whatever is up in there that is causing the irritation gets flushed out, it may stop?
Whereas I wonder if blocking it could lead to a sinus infection? Your body is trying to help you, so it is generally best to help rather than hinder that process, unless you really truly do know better - though speaking of, I am not a doctor, it’s just thoughts.:-)