BumpyCamper-The furring strips in the shorter cabinet are free standing. I added, I don't know what to call them, wedges, for lack of a better (or more professional) term. The unit sits all the way to the back of the cabinet and I cut a few more pieces of the furring strip to wedge, and gorilla glue, between the front of the frame and the front of the cabinet. If the glue doesn't hold long term, putting a screw into the brace shouldn't be too hard.
The taller cabinet had one shelf in it. What was I supposed to do with one shelf above my head? I removed the shelf, but found the supports were well connected to the wall, so I incorporated them into my design. I used them to hold the frame from the top. I drilled pilot holes and then dropped screws into the frame. It feels rock solid. So far the shelves have held up well, but it's only been one month and two short trips. If you have further questions, feel free to PM me. I can give you further specifics without boring everyone else.