Insulation, Radiant and Vapor Barriers - New tounhouse: insulation between common walls

I live in VA in an end-unit townhouse built in 2003. The inner walls (=common with the next unit) feel cold during winter and cold air is coming from the outlets, etc. Does it make sense to put insulation b/w the drywall and the common wall (thick sheetrock? not sure what it is) in the finished garage and the ground floor room that is behind it? What kind of insulation can be used?

I'm also thinking about insulation for common wall on other floors, mostly for soundproofing. Any comments?

Insulation isn't designed to stop cold air. It's designed to hold heat in a house that has been properly sealed.

So maybe you need to consider having an energy auditor find your major leaks, or retrofitting with an air-sealing insulation.