Here's the result after $20 of wire shelving and some organizing time:
I bought a set of white wire stacking shelves for each shelf so that I could stack glasses without the danger of them falling over. I stacked the large glass tea mugs and short drinking glasses on one shelf on the bottom level. Then coffee mugs double stacked in the middle level. And then the top shelf has our less frequently used small tea glasses and turkish coffee cups, contained in plastic bins and stacked on wire shelves.
These wire shelves made it so I can fit all of my glassware in the cabinet all at the same time - neatly and without unstable stacking. These were a great purchase! I got one at Fred Meyer and two from Walmart and all of them were pretty reasonably priced.