I don't know what happened, but after soaking it the second time, the wheel would not turn at all. Using the pictures so kindly provided in this thread, I took it apart, removed the wheel assembly, and got it to turn by hand with the help of pliers and a crescent wrench.
I applied some "Break Free" (teflon oil gun lubricant) and ran it back and forth a few more times, each time it got easier and easier. After a few more cycles, and the application of some more lube, I wiped off the excess lube, and re-assembled the airbrush. It now works like new! (I'm not sure if the oil will come out with the paint or not yet as I have not yet used it to paint anything..)
Has anybody ever suggested lubing the moving parts on the inside of an airbrush? If not, is there a reason why? For example, the oil mixing with the paint, etc? Anyway, for now the stickiness of the wheel assembly seems to ahve been cured. I'm worried that the next time I clean it out, the lube will come with it and I'll be back to square one. So, I might be looking for a new, all metal airbrush that won't bind up like this.