Yeah, 16" centers is more normal - so you have 1" sheet rock plus 1/2" panels on either side of 2x6 studs on 16" centers - should be a pretty solid wall already.
Here's what I would do in your case -
1. call some insulation contractors and get estimates on having the entire wall insulated with blown-in cellulose. They will have to cut a couple holes in each stud cavity (every 16") - blown-in cellulose reportedly has a pretty high sound isolation capability, so this will help several dB, probably around 8-10 just for that.
2. Patch all the holes (or have them do it, if it's part of the deal) - make sure the patches are airtight and solid.
3. Ask the same contractor if they sell acoustic rated caulk - don't accept any substitutes. There are several brands, such as USG, Owens Corning, etc - Thoroughly caulk every crack, corner, seam, etc, in the existing wall.
4. Find a sale on sheet rock and buy enough to put 2-3 more layers of sheet rock over the existing wall - different thicknesses for each layer will help also. The way you attach these layers is CRITICAL - NONE of the new layers should be fastened directly through the existing wallboard and into the studs. Period. Ever. You should use the studfinder to mark where NOT to put any fasteners for these new layers - I use masking tape on floor and ceiling to mark things like this. You can write on the tape what it is, and whether to fasten there or to NOT fasten there.
More 4. You want to offset ALL joints from layer to layer, so that there are no coincident cracks in the new layers. In this case, you would want edges of panels to fall about 2-3" to one side of a stud. This is where a line of Laminating Screws should go, on 12" centers vertically. (You follow a stud with screws, staying about 2-3" to one side of the stud, and spacing screws 12" apart vertically. It would be a good idea to spread a stripe of joint compound vertically where the rows of fasteners go, this will strengthen the bond between layers. The type of screws you want are called Laminating screws, or as USG calls them, Type G (for Gypsum) - they are designed to fasten layers of sheet rock together WITHOUT fastening to studs. This is VERY important. These screws are not very commonly stocked in most places, you may have to special order them if your local drywall contractors don't carry them.
Even more 4. You want to use 1/8" spacers under each layer of wallboard when mounting it to the wall - once the joint compound and laminating screws are in, you pull the spacers and caulk. Then add the next layer. If you can afford it, I'd go with 5/8, then 3/8, then 1/2" sheet rock.
Basically, what you want to end up with is all stud cavities filled with cellulose, the wall patched, then 3 layers of dissimilar thickness sheet rock over that, with NONE of the new layers fastened directly to the studs. Their only fastening should be a glue line running parallel to and about 2-3" away from studs, beefed up with laminating screws that ONLY mount each new layer to the old. This means progressively longer screws for each layer. What you DON'T want is total joint compound coverage between layers, only a line 3-4" wide where the fasteners hold the new layers to the old. You want each individual, different thickness layer to still have its OWN RESONANT FREQUENCY. This improves Transmission Loss through the wall. NOT having any fasteners holding the NEW layers onto the old by going into STUDS, cuts flanking noise and improves the wall performance by another 8-9 dB typically. The extra mass of 1-1/2" of drywall will improve your side of the wall by another 6-8 dB, and the cellulose should give another 8, plus now you have different mass in the two leaves of the wall, so you would probably see a total improvement of around 18-20 dB. This might be even larger if there are already cracks and holes that sound is getting through, if you do a super-thorough job of caulking with the REAL STUFF.
If you have this done by a drywall contractor, do NOT, repeat NOT, let him talk you into using Resilient Channel or "furring out" the new drywall. All either of those will do is to WORSEN the sound isolation. Cruise the other posts here on wall construction and you'll find that a TOTAL of two leaves (centers of mass) and only ONE air space is the best possible use of materials for sound proofing. Period.
If you decide to DIY this, I would recommend a good book on drywall construction - there are far too many details of do/don't for me to get into here, I've tried to limit details to those that are DIFFERENT from normal construction.
Man, if this post doesn't elicit more questions, you obviously weren't paying attention...

Steve