Flannel works well for the body of the dipe, but IME, it makes a lousy soaker. Terry (an old towel) or cotton fleece (an old cotton sweatshirt) work far better as the soaker.
You cannot make a fitted without elastic. If you are truly that afraid of elastic, then I'd skip fitteds and just use the blankets as flats.
I'll tell you the easiest and least headache-inducing way to put elastic into a dipe though... Decide where you want the elastic (i like legs and back), topstitch about 3/4 inch in along that section only (a little wider than your elastic - 3/4 in is good for 5/8 in elastic) stopping and starting where you want the elastic to start/stop. You just made a casing. Decide how long you need the piece of elastic to be (relaxed it should fit snugly around baby's thigh), and using a sharpie mark that length on the uncut elastic. Thread the elastic in the casing, make sure its not twisted and sew a line from the outside edge of the dipe to the previous line of stitching, catching the loose end of elastic in your stitches. Sew over this at least 3x. Pull the other end of the elastic until you can see your mark. Line it up, pin it in Place and sew the other end of the casing the same way. Now you can cut your elastic. Repeat for the other casings. You never have to stitch along taut elastic, which is a pain.
Another option is buttonhole elastic, which is wonderfully adjustable, uses similar casing setup, but you have to set in buttons in a way they wont tear out yet are still accesible.
Google for a free pattern, Ritas Rump Pockets are one ive heard of, it should have basic instructions with it. Or, easier would be buying a kit from Diaper Kits, so you can do one completely before attempting your own design.