Does 100 yr old concrete floor need replacing?
Concrete is porous, regardless of whether it's new or old. If you replaced it with a new floor, you could have a drain installed around the perimeter to stop most of the moisture infiltration. The best solution is to install a barrier around the perimeter of the exterior, which requires a lot of digging to get to the bottom of the foundation and installing a drain at the bottom to remove the moisture that gets to this barrier. A less intrusive solution is to install a French drain anywhere you have moisture problems around the outside of your home.

Regardless of how you decide to fix this problem, you should ensure that the yard is properly graded away from your property on all sides so that any moisture in the ground goes somewhere other than against the concrete foundation. This, combined with your work on the gutters, could be enough to make the moisture levels tolerable.