Lo and behold, there was a photo incidentally showing some of her hand basting---with really short stitches. Why didn't I think of that myself?! Yes, it will take more time than the traditional long stitches I've always seen, but it will hold the layers better and the stitches won't be nearly as likely to get caught on things while machine quilting. This just might work! The other downside is, of course, picking out the threads from under the machine quilting, but I'm willing to deal with that, too. Thanks, Tonya!
I'm even getting excited about the prospect of marking some simple designs and trying to follow the lines. Hey, I have a new machine now, with a giant 10-inch bed, and my first attempts at some machine quilting on it were encouraging. I think the way the foot works keeps the quilt more stable than on the old machine.