I had the exact same problem when I bought a new house. I used something
called "Tile Grout Coating," which I found at the hardware store.
It coats over the grout stain. I put it on about six months ago and it
still looks fine (I didn't use it in the shower part yet, but the rest of
the bathroom gets humid and a little wet too, with no ill effect). I was
kind of surprised it worked.