The partition wall is two separate 2x4 walls, with independent sole plates, and offset studs. That part is covered.
I'm pretty much stuck with the dimensions of the existing wall.
Measure twice, cut three times, start over. Repeat as necessary.
The separate stud walls with an airspace will help the most. Insulating both walls will help with sound attenuation. Hanging the sheetrock on Resilient Channels (RC-Channel) will help as will using 5/8" or a couple layers of 5/8" Sheetrock. From most of the studies I have seen adding any septum sheathing (soundboard, plywood, or sheetrock) in the airspace between the walls will decrease sound attenuation. Keep the sheetrock off the floor and use some sound insulating caulking. There are some decent products out that can help; homasote, quiet rock, etc.. The biggest benefits from those products are really following the manufacturer’s installation instructions. You can get fairly close to those products (without the added expense) with properly installed sheetrock. Some of those products perform really well like the Quietrock 545, but it's not cheap or easily obtained.
The Gypsum Association manual GA-600 http://www.gypsum.org/download.html has STC ratings for many assemblies. Section III has some basic sound control information. We use it quite a bit when putting together assemblies for our buildings. Many of the Sheetrock manufacturers have similar assemblies and sound test info. The USG design studio is pretty good: http://www.usgdesignstudio.com/wallselector.asp
I'm with Kevin. To dampen the vibrations that transmit sound, you want an energy absorbent material that is not attached to anything. The high end home theaters make a wall where every other stud is offset by a half thickness. Then a relatively thick dense sheet material (something like a thick dense tar paper) is snaked through the structure and just hangs there. No issues with gas lines etc because it's not completely filledDo some searching on home theater. Filling with foam or insulation isn't as effective. Although it's readily available material....
two layers of 5/8" thick x fire rated sheet rock (x-rock) works wonderfully. use glue and srews.
Regardless of the products involved, the principles behind what you need to do are: de-couple the opposing wall surfaces (reduce the "drum" effect), seal the rooms from each other (if air can leak, so can sound), and add mass (takes much more energy to move more mass).
I wish this forum software allowed voting on people's comments. +1 Jason. It is always good to understand the why before attempting the how.