I agree with you about the IKEA KALLAX shelving units. If you push on them from one side, the whole unit bends! If I was forced to use the KALLAX, I would install support bars across the rear. The KALLAX is also a little too deep for my liking, though some people view that depth as a plus. I also don’t like that it has an open back, which allows dust to enter the interior of the cubes.
IKEA offers another storage unit that I CAN recommend: the EKET. This model is available as a single 13-3/4" cube (interior 12-3/4" in width, depth, and height, just barely big enough for LP covers), a 2-cube model, and in what I consider the best choice for LP’s, a 4-cube model measuring 27-1/2" in width and height, 13-3/4" deep. The interior of each cube is 12-3/4" in weight, width, and depth, just big enough for "normal" LP covers---that is, all but MoFi One-Steps, Analogue Productions UHQR’s, etc. The EKET has a rear panel, which not only prevents the entry of dust into the cubes, but also provides structural integrity to the assembled unit (it comes as a flat pack, very easy to assemble). The rear of the EKET also sports holes drilled for support braces which may be installed when stacking multiple units, making multiples feel like a single unit.
I have fifteen EKET’s, three high and five wide. The rear of the EKET also has holes drilled for adjustable plastic support braces, for securing the unit to the wall it is placed in front of. You put the EKET (or EKET stacks) where you want it/them, and unscrew the braces (real left and right) until they rest against the wall. My stack of 15 EKET’s are completely full, and are absolutely unmovable. The assembled stack feels like it’s a built-in!
IKEA sells the 4-cube EKET for $60, and offers it in brown walnut (which looks cheap and pretty ugly), white (again, cheap), and dark gray (by far the best looking). I bought my 4-cube EKET’s when they were priced at $50, so a 3 x 5 stack of 15 cost me $750. More than the KALLAX, but in my opinion far superior and worth the difference in price. IKEA also offers a pack of four short adjustable wooden legs for the EKET, which raises the unit off the floor by a few inches. Great for preventing dust from migrating from your rug/carpet/floor into the cubes.