Cheapest way to screw in drywall close to a big hole without patching the hole
My cheap solution for small patchups, is to use a 6 or 7 inch piece of a paint stirrer stick.
Smear it with wood glue or construction adhesive and center it on your hole, inside the drywall, aimed where you might want to drive your screws.  A screw in the middle can be a handle.
You'll have to keep it place while it dries.  If you don't care about more holes, a drywall screw on either side will work.  Remove and patch when done.  Predrill though, as the sticks will split.   Or tie some string around the middle, and use another stick on the outside to hold it tight.