I agree a damp basement shouldn't necessarily cause the floor joists to rot. Any sign of termites or ants?
Small tunnels or channels or borings or tiny piles of saw dust here and there?
I would first want to know exactly what the foundation walls are sitting on, if anything. You would have to dig down on the inside along an outside wall to determine exactly how the walls were constructed.
The fact that the timber joists are sitting in pockets may also be what is holding the walls in vertical position after all this time. Wholesale removal of the timber joists may cause a wall collapse.
How's that for a start? I know...scary! I'm assuming you guys have deep-enough pockets to take on a structure of this kind and age.
Replacing the floor joists (per se) shouldn't be a major issue. You will need an alignment plan of course.
How about some pictures?