Yes that is definitely do-able. Once the floor is dry, get some of the Rot Doctor CPES as mentioned above. Saturate the end grain of the OSB to seal it up, then you can sand or plane down the bottom to reduce the thickness of the OSB and keep it from dragging. The Rot Doctor CPES is actually an penetrating epoxy so it will restore strength to the OSB, plus since you had wood installed over top of that it should be plenty strong.
Once the OSB is the proper thickness, you can actually paint that CPES on the bottom of the OSB to seal it up. With a drip edge installed, that should fix your problems. I would paint the whole underside of the floor and the other edge as well if you have any questions about whether it might be in for the same damage.
You are significantly ahead of where I am....the factory installed a sheet of Reflectix (think mylar-coated bubble wrap) under the carpet instead of padding to help with insulation. That Reflectix sheet kept the water from soaking into the carpet, so I never knew it was leaking. Meanwhile the Reflectix also didn't let the wood breathe, so that's black mold that you see in the picture...a real mess and why I am replacing the whole floor.
