Wayne,
Screws being a problem in plywood is true with wood screws, which from some reason are so conical. I used to drill pilot holes through both pieces of wood, make the top panel hole larger so the screw didn't catch in it (let's it be clamped down much better), countersink that, put in a dowel plug when done. Now, with the new thin drywall screws, don't really need the pilot hole in the sideways/bottom piece. Much better clamping, too. The best screws now are Spax, made in Germany, not too expensive, designed for wood, but kinda like a drywall screw, have *really sharp* serrated edges on threads, need no pilot holes- just line it up very straight and clamps like a demon.
The ply at Home Depot with the birch or oak outside layer is very good. This new stuff they have, called Sandex or something like that, is not.