What you suggest would help, but an even better solution (if you have room) would be to add another set of studs from floor to ceiling that do not touch the existing wall. The studs could even be installed sideways to minimize the space taken up. Then install fiberglass insulation (do not compress it too much) and follow up with 5/8 drywall and finishing.

Eliminating the contact between the two walls will help greatly.

Good Luck!