How to prevent racking on a large free standing wardrobe

As Chris Cudmore states in his comment, it is most likely a leveling issue (the other, but less likely cause could be misaligned hinges).

Place a level across the top of the front of the unit. Note whether it is level or not. Then do the same across the top of one of the sides.

If the unit is tipped, but only front to back (the front test was level), you should try to raise both of the legs on the low side. Any solid lift that fits under the leg base will do, but the simplist approach is usually small plastic wedges.

Similarly, if the unit is only tipped to one side (front test shows slant, but side test does not), wedge both legs on the short side.

If there is a tilt both side to side and front to back, try wedging just the lowest corner. Then add additional wedges as needed. You might need three, but not four.

Keep testing in both directions until the unit is level all around. Check the door again. It is likely that both doors may swing only when pushed.

P.S. The reason only one door swings shut now is probably because the swinging door is actually falling toward the low side. The opposite door is already on the low side when it is opened. Minor differences in hinge tension may make one more likely to swing than the other, but the first line of defense is to eliminate the reason for the swing rather than suppressing the ability to swing.

If this does not cure it, you may have a hinge problem, which is another answer altogether.

SUPPLEMENT: On many more recently manufactured furniture, there are often built in levelers that avoid the need for wedges.

These are threaded pads set into the bottoms of the legs of furniture. You raise the height of the corner by twisting clockwise (when looking from the top). If you have them, use them.

Also, once you determine the height to raise a corner, you may be able substitute rubber furniture cups, if the plastic wedges seem too slippery. These will work if the height is about the amount that you have determined you need using the wedge (or if they can be trimmed to make it right).

Plywood will not sag over time if you intend to use it for the door faces. I recommend birch plywood for not only its hardness but looks. However your hinges and what they are attached to are the big concern in this project. It looks like you've used a low grade pine to build the unit. This is a soft wood and a couple of cabinet hinges screwed into this are going to loosen quickly. Your best chance for longevity is to use a piano hinge so that you have a continuous hinge from top to bottom with 15 screws or more on each hinge. I highly recommend ordering a stainless steel or flat nickel piano hinge for this. What they have in stock at the box stores is of the "bright" variety and looks cheap to me.

I'd also recommend that you not attempt to build the door to be perfectly flush with the "lip" at the bottom. Leave yourself 1/4 of an inch gap. Units like these always sag at least to some degree and you wind up with the door scratching the "lip" and its going to look really bad with time.

Also take a look at ball catches to use either at the top or bottom of the doors to hold them in place once closed.