After weeks of procrastination I finally unscrewed the trap thing under the sink. Some absolutely vile water drained out. The trap was worrying clear, with only some bicarb soda, which I cleared out. Screwed it back up. The rubber o-ring thing broke while I was working, suspect it's going to leak now. Anyway, it's still not draining properly. I don't know what to do next, may have to call a plumber.