So just my own experience... I have tried saving dried grass clippings in the same way I've stored the dried leaves, and it gets really dusty no matter how much I dry it out. Could be that it molds at some point and then the mold dies off and that's the powder left behind, or just something about the dry decomposition that makes it dusty, but either way, dried grass clippings don't keep the same way leaves do, so I'd strictly do leaves if you're planning on saving them for months.
Make sure the leaves are surface dry or mostly surface dry (I've gone as far as to spread them out on the lawn to dry after a rainstorm), and pack them loosely so air can circulate in the bags and they can finish drying out during storage.
