Well Schaffhauserstrasse is unfortunately in Zürich, not in Schaffhausen.
There is this place
across the river in Feuerthalen. They seem to have "nice" as well as "fancy" dress... There must be a few more places. I grew up out there and while I never hired outfits myself, friends of mine regularly did... Before they had cars, so they cant have gone all too far!
Tried the yellowpages or search.ch? The term you'e probably looking for is "Fasnacht" or "Fasnachtsartikel" - that seems to b e the main occasions for locals to get dressed up. Across the river the local event is called "Hilari" by the way, their version of Fasnacht. Try searching for that...