I thought about why the grass clippings didn't just go flying all over the yard like the sheets of paper did. Well, grass is wet (making it heavier for longer) and it kind of interlocks over itself so that when it dries it makes a kind of mat. But, it's early April and there is no grass from which to get grass clippings. And, I need mulch now!
Then it hit me: I could fake grass clippings with shredded newspaper, if I got it wet. So, I shredded a stack of papers (about three inches high) to get about a kitchen trash bag's worth of shreds. Then, I got it wet by pouring water on them and tossing them around to get soaked.
Once I had my "mulch" I put it on the garden. I tried a very small area first and it seemed to do okay. That is, it didn't fly around or turn into confetti all over the yard, and it seemed to keep the dirt underneath moist. So, I did the above amount today. We'll see how that goes, but I'm hopeful. Grass clippings will be better, but these will probably do until I can get some. Also, I'm not sure how the ink in the paper will affect things; it may or may not be bad for the soil, I just don't know. I only used newspaper, not that glossy stuff they use for ad fliers, so it's probably okay.