The solution to all the sagging and unsupported floors was to create a new beam that was strong enough to hold up the 1st and 2nd floors while also providing a place to attach floor joists to.
The trick was, we could not remove the existing 8X8 timber as it was still solid on its south end and was supporting the staircase.  It also was built into the stone foundation.

I hired my friend Robbie to help me and we opted to create another beam which would be placed along side the old beam, and slowly we would transfer all the structure from the old to the new.  The new beam was created from four pressure treated 2 X 12's  glued, screwed, and bolted together.  This beam would span twenty feet across and be perched atop two cinderblock pillars constructed on top of 18" deep concrete footings.

Here is a couple of shots of the footings that I poured and the pillars that Rob built: