Field-stone Foundation on Top of Bedrock, Keep the Water Out?
Not sure about inside, maybe damp rid or a dehumidifier. I don't suppose it wouldn't hurt to pack concrete in that gap under the wall/foundation (but you should dig this out from outside first).

The truth is, water coming in must be taken care of from the outside. A picture of the outside might be helpful.

I understand that you are on bedrock, but just to be sure, whatever soil exists outside should be graded away from the house. If that's not possible (or is not helpful) then you need to (dig out the outside and) seal the outside of the foundation. Tar and rubber liners are good for houses built into the side of a mountain. Again, packing concrete in that gap under the wall/foundation would probably be helpful. French drains should be at the foundation level... so that sounds like work (cutting into rock?) for you. If this is in the side of a mountain or hill, a cement patio or sidewalk (sloping away from the house) with a retaining wall for the soil would be the most effective solution (for worst case scenarios).