Join Date: Apr 2010
Location: Near Boston, Massachusetts
First: you can absolutely make a dado with a backsaw and a chisel, rather than a router plane or router. It just takes more care and time. Make the side cuts carefully, and use a knife to mark the depth. Start using a chisel to run across the grain, removing triangular sections of waste, so you wind up with a "W" sort of shape. Once you're down enough that it gets awkward, set the chisel into the central triangle of waste, bevel down, and start removing waste that way. With the bevel down you'll have a lot of control, and shouldn't be at much risk of going too deep. For the final passes, come in from both ends of the dado, to avoid blowing out the wood at the exit.
Second: you shouldn't need to clean up once you're done cutting. The sanding is why you're getting rounded corners, unfortunately. If you cut a dado across the grain and you need to make it slightly larger, use a chisel instead of sandpaper to widen it. Ideally you'd use a chisel wider than the material, but if that's not available use what you've got. In your current situation, I'd probably wait for the plane to arrive: fixing those corners freehand with a chisel could be tough.