I complained to my ophthalmologist about my tearing eyes. So annoying. I was told about the same blocked drains. The tear ducts worked fine producing sufficient natural tears, but there are literally drain holes I n the lower lid area that can be blocked, then the tears overflow and the eye streams. Right in the office exam chair he used an instrument, perhaps a thin wire or needle, to clear the drains. After the puncture procedure he used a type of syringe to force-flush the drains. I could feel the tiny stream of water running at the side of my nose (inside) right to the back of my throat. It was unusual but not painful. None of it was painful and it took barely a minute each to unclog and flush the drain in each eye, one at a time.
The difference was amazing. All winter I noticed no tearing and nuisance wetness and blurring. Now that it's summer, I'm out in the breeze going sailing almost every day. No tears!
I wasn't given any fancy name for the procedure and I felt no discomfort, maybe a tiny pinching feel during the "drain clearing" and the odd drip feel as the flush followed to clear the drain completely. It worked, it was quick, there were no tubes inserted in my case. Perhaps if simple drain clearing didn't work orvrecurred he'd try tubes. But no so far. I'd follow the doc's advice and let it get fixed. It will give much greater comfort.
Medicare covered whatever code was used for the billing so I guess this is pretty standard. Hope it works for you. Don't be scared.