I've used a lot of Homosote for sound deadening. I wouldn't use it for a finished wall (it's basically compressed newspaper- very soft, would ding easily, and extremely absorbent, so it would wick up dampness from just about anything).
Is there a reason you don't want to use R-13? It would help an awful lot. If not, can you increase the density of the wall some other way? (stack bricks, sheetrock scraps, plaster scraps, etc?)
There are two main characteristics to reducing sound transmission- increasing density and mass, and de-coupling the components to break the vibration paths.
That means, try to get the wall surface panel (sheetrock) on one side of the wall physically disconnected from the other. So, yes, staggered studs would help. They don't have to be 6". They could be 2x3's (if not load bearing). Or resilient metal channels ("hat" channel) run horizontally.
If you can't re-frame, you can get some separation by padding one side of the studs with strips of padding (homosote would be great, but thick cardboard might work too). Use continuous beads of construction adhesive (Liquid Nails) on the studs, and screw your sheetrock using minimal screws. I've built some walls where we actually removed the screws after the glue dried, so the screws wouldn't transmit the sound.
Different frequencies are transmitted through different materials. Most of the sounddeadening walls I've built use multiple types of density materials to reduce the transmission of different frequencies (after de-coupling the framing, and adding density with insulation). So, 1/2" sheetrock will keep out different sounds than 5/8", or 1/4". I've built walls that had 1/2" on one side, than a double layer on the other side (5/8" and 1/4").
I'm not an acoustical engineer, but I've worked for several.
