Wrap the walls with roofing felt (tar paper) or building wrap, then add siding or a layer of thin rough sawn plywood to cover it all. You can use "Z" flashing if you can't find sheets tall enough to reach from top to bottom.
Another option... Unscrew the plywood from the studs, add 3/4" furring strips to the studs, then reattach the plywood with longer screws. Brace the building and you can do one or two sheets at a time. This would let you extend the plywood over the floor structure to keep out leaks, then add "Z" flashing at the top and add a strip of plywood to make up the difference.
If there's only 12 screws in each sheet, it almost sounds easier to dismantle the shed for the supplies, then rebuild it the way it should have been. Assuming the rest of the shed has so few fasteners, it should come apart with minimal damage to the materials.