If you are not averse to designing/building the cabinets yourself, it is not a terribly difficult project to execute. You don't even need a wood shop, only a few select tools.
Seek out a local full service lumberyard or specialty wood shop (not a big box store, a real lumberyard). If you bring a cut list of panels that you need, they will usually charge only a nominal fee to break down the plywood sheets for you. This will save you from trying to accurately cut all of the large sheet stock (if you do not have the capacity to do so yourself).
At the lumber yard, you can also purchase dimensional stock to produce the face frames for the cabinets.
At this point, all you should need is a miter saw to break down the wood for the frames, a pocket hole jig for the assembly, and the hardware/hinges/etc.
When it comes to the doors, you have a couple of choices. You can follow the same kind of pattern above (purchase dimensional stock, buy an entry level router / rail and stile bit / edging bit, have the lumber yard break down the inner panels for you). Alternatively, you can order unfinished custom cabinet doors online. All you need is the dimensions for the doors, and then you can just choose the style you want.
Of course, this will take a significant effort to design the cabinets specifically for your space, and it will take some time for you to assemble them and work out all the issues. It all really depends on your schedule vs. how much you want to spend. Doing it all yourself will save you money, but will take a good bit longer than hiring the job out or grabbing off-the-shelf cabinets.
Best Answer
The back of the cabinet plays a large role keeping the unit square. My suggestion is to forget about trying to re-work that cabinet. There a bunch of concerns:
In the end the amount of messing around with it highly likely outweighs any benefit of trying to use this cabinet for another purpose. Instead invest in something that is already a wall mount shelf unit.