Help! My Bathroom Sink Won't Drain and Smells Awful!

I've got a frustrating situation with my upstairs bathroom sink. It often either won't drain or it gives off a terrible smell at the same time. I've tried various fixes: chemical cleaners like Drain-O, boiling water, and even vinegar with baking soda, but nothing seems to work long-term. The only temporary solution I've found is removing the tailpiece and p-trap to clean them out with a garden hose, which reveals a lot of nasty, smelly gunk. We switched to a metal tailpiece because the plastic one my wife picked seemed to attract even more gunk, but the smell came back quickly. After being away for a month, we returned to find the sink draining at a glacial pace. We're careful about what goes down the drain, using natural toothpaste and soap, but I'm still stumped. I'm considering getting a hand drain snake to explore further down the pipe myself, but I'm not a plumber and don't want to make things worse. Any insights?