My first attempt, the holes didn't line up, because I'd allowed the thing to move in between hole-drillings. I thought about it, frustrated, for awhile. I was getting hot and sweaty from my efforts, so I stripped off my shirt. (What? Guys can do it! And I have a "privacy window" in my bathroom. But even so, I felt like I was involved in some kind of kinky home-improvement porn. Oh well, I was more comfortable. (No, I didn't take any pictures.)) Then I had some diet coke, and I came up with the solution to my lining-up problem.
I picked the upper-right hole I'd made, and called it correct. I installed a plastic anchor in that hole, put the cabinet back up, and screwed it into the anchor. Once it was semi-securely attached to the wall, it was much steadier. I was able to level it again and drill the other three holes much more easily. Then I let the cabinet pivot back and forth on its single screw so that I could install the other three wall anchors in the holes I'd drilled. I then leveled it again, found that the holes were still lined up properly this time, and installed the other three screws.
Success! The cabinet was securely fastened to the wall! I'm pleased with my little endeavor. It looks pretty nice too, although it's still empty right now.