Three sets of wires and no clear way to ground
If the box is grounded, look around for a hole in the box that's tapped for a 10-32 screw.   It may be on a little "hill" and already have a screw in it.  Attaching a solid-core wire to this, or a wire with a spade or ring terminal, is the right way to "grab" ground off the box.  I like ring terminals, they're a little harder to work with, but you can put more than one on a screw.