A touring bike will do fine for just about any other kind of riding except for serious road racing or rough off road stuff. It will handle anything the hybrid can and still be smoother and faster on the pavement.
You'll be shopping used for that price. Shouldn't be that difficult to find something good. There usually isn't any difference between serious bikes for men and women, at least not at your size. Some are designed a bit differently for women but the difference is subtle.