If you need to use a bathroom, definitely when out and about (restaurant, station, etc) you'd ask for "et toilet."
Really the only situations where I see "et badeværelse" is when talking about the rooms in a house / apartment. As in "Der er et stort badeværelse til venstre." So not a thing that tourists would normally encounter.
you shouldn't say "sorry" but "excuse me". sorry expresses regret while escuse me is to attract someone attention. once again I'm here to learn Danish but I don't want to mess up my English. exercises should be prepared by linguists who have perfect knowledge of both languages.
But as has already been stated, badeværelse is the room with a bath in it, not the toilet. Granted, in many houses they are in the same room, but Danish, unlike American English, does not use "bathroom" for "toilet".