Musty smell is gone, have no idea what it was but a very light mopping of bleach got rid of it completely. I am lucky on that front.
The cardboard boxes, I should have said to avoid misleading, are in a separate central part of the garage, dry and off the ground.
I like the idea of putting a protective coating on these "sentimental" boxes, since replacing them with plastic doesn't appeal to me.
I have plastic tubs in several other places (including under the sink where they are the right thing). But these boxes, I think are going to stay.