How to resolve uneven meeting of new and existing drywall pieces
Cut a few furring strips 4" longer than the opening. Place them behind the opening and screw them in place through the existing drywall on each side of the opening. Now place the new piece of drywall in the opening against the furring strips and use drywall screws to fasten it to the furring strips. you should now have a smooth surface to mud, tape and paint.