The balance we are trying to keep in the compost pile is sometimes as a shorthand called wet vs dry, but it is not. The balance is Carbon heavy vs Nitrogen heavy. So the Carbon heavy stuff (aka "browns") like leaves, paper, corn cobs, straw, tends to be harder, drier. The Nitrogen heavy stuff (aka "greens") like pulled weeds, most kitchen scraps, grass clippings, tends to be softer, moister. But that is internal moisture (water content) we are talking about, not what is on the surface But it is the C:N ratio that is important. If you are putting fall leaves in your pile, you are adding browns, whether or not the surface of the leaves has gotten wet.
You might want to read the intro to composting threads at the beginning of this Composting Forum. In one of them there is this discussion about C:N ratios viewtopic.php?f=35&t=29022&p=157417&hil ... io#p157417
Grass vs dried grass is also not an important distinction except that wet grass clippings can mat down too much for good mixing and air circulation. But if you have lawn grass clippings and they dry out, their chemical composition has not changed much. There is
a distinction about WHEN the grass, leaves, grain stems, etc leave the plant. Green tree leaves in summer are a "green," higher in N. The reason that the same leaves when fallen in autumn are a brown is that the tree thriftily sucks the Nitrogen back from them before releasing them. The same is true of hay vs straw. Hay is grain stems that are cut while still green and growing, right after the grain is harvested. It still has a lot of Nitrogen. Straw is the same stems that are left standing until they dry out naturally. The plant has used up the Nitrogen out of them, so it is a brown.
If your leaves are tending to mat, just tossing them or mixing them in the pile should take care of that.