Happyman, you have a good system. Thats a good way to do it.
I had a 3 bay system at my last place, smaller than yours, but same idea, New stuff, Cooking, and Ready Stuff.
I have the long thermometer too, that helps. Moisture can be a long conversation, depends on area, time of year, and raw materials.
I need more water for dry fall leaves, and less in spring time when the grass and manure has it's own moisture.
When my pile needs more moisture, I use the FEL to flatten and hollow the top of the pile, so it will catch more rain. If I think it's wet enough, I make it pyramid shaped. A rain barrel is good thing to dump a bunch of water directly into the pile. or even a trash can full.
As far as breaking up the clumps, well, my wild guess is that the clumps are parts that haven't totally decomposed. If you have a way to screen it, I'd put the clumps back in to the Cooking pile, and maybe they will break down further??
You can build a simple gravity screener with 2x4s and wire mesh. Set it at a steep enough angle, and drop the material down the screen, clumps and rocks will roll off down the end, the fines will go through the screen and make a nice clean pile.
Over in the Build It section I posted a "Down and Dirty Screener" thread a while back. Construction is really easy.