Do I need to cut into my drywall to install a switch?
You're going to have to do drywall repair if you move this switch.

So since you HAVE to do drywall repair regardless, why shy away from doing it the right way?

Cut away a big enough piece of drywall so that you have full access to the existing box, the space where the new box will be, and keep cutting right and left until you hit the bordering studs on either side.

Do your electrical work.

Cut 1 piece of drywall to fit the huge gaping hole you made.  Cut out space for your new box.

Cut pieces of 1x2 to fit vertically on the insides of the left and right studs, and to fit across the tops and bottoms of the opening.  (6 pieces total)

Install the vertical pieces parallel to the studs with 1" side facing you, so you have a screw edge.  Install the horizontal pieces so that the 2" side is facing you and half of it is behind the existing drywall.  Screw it in place in the existing drywall.  

Take your cut piece of drywall and mount it on the 6 1x2"s you just installed.

Tape and mud - 3 coats - fill coat, feather, final.