All the wood isn't soaked. It'd take a flood to soak those stacks. There's a lot of bad info in this thread. OSB (Oriented Strand Board) isn't what it used to be. It's much better. I'd still prefer a quality ply but prices rule that out for 98% of us. OSB's vapor permeability is very good as well as it's drying ability, many think it's better than ply for that reason (as well as weight). Left in "bulk water" OSB is done for. However, bulk water isn't kind to most woods. The issue for extremely wet OSB is diminished strength and future mold.
I wouldn't worry about either for those top sheets.
Anyhow, back on those stacks. I'd pull the first (top) sheet and get working with the dry stuff below it. Then, by the time you need it, it's dry and easier to lift/cut/etc. Even the way it's pictured, it's probably not that bad as it sits.
Oh, and to answer your question, yes, cover them up with a tarp when it's raining. If you get some sun, uncover it if it was wet previously. The contractor will appreciate it.
Experience: 10+ years framing/rough carpentry