The 2 studs provide excellent main support, and you can use 1/2-in plywoord or other dimensional lumber pieces like left-over "1x4" for support at the drywall transitions.
Backing does not have to run the full length of the seams. In your case, if about 3/4 of the seams are backed that's fine. First screw the backing to the existing sheets, half-on, holding the backing firmly against the back of the drywall. You'll need a screw perhaps every 6in. Then when you apply the patch piece you know where there is backing based on the position of those screws.
Some places sell half-sheets, which with your 32in width and probably less than 48in height should be perfect and easy to transport. You may also be able to get some free plywood strips or 1x4 cuts for the support.
Straighten your cuts and stay at least an inch away form the corner. How straight? A gap up to perhaps 1/4in is totally fine. In fact I find that gaps allow mud to join the edges better, than when drywall tightly butts.
You can cut an "L" as your cutout, or replace a whole 32x48 rectangle. Depends on whether you want to redo your baseboard.
Clean-up the existing edges by bevelling them like a V on both sides (where old butts new) and thus shaving off any loose paper at the tattered edges. After installing you can use mesh tape.
As mentioned in the comments, consider Rockwool noise reduction, and any horizontal support brace (e.g. 2x6 / 2x8 on edge) for a hanging/suspended cabinet before you apply the sheets.