Over here, at least in my parent's generation, a trollop or strumpet would have negative conotations, thus so would doxy, esp. if we are equating it with prostitute, which gets shortened to "prossie." Except for the latter, they all seem to be old-fashioned words, which I think of in more use when my parents were growing up than now.

The other word which comes to mind is "moll" as in a girlfriend of a mobster being known as a "gun moll." Not quite a prossie, but definitely a woman of loose morals in that she was unmarried and having sex with the guy.:-)