Will supported 2x8s or unsupported 2x12s added to existing joists result in higher rigidity?
Side answer - beam stiffness goes as depth cubed, width directly.

12 cubed is 1726, 8 cubed is 512

So 2x12 is 1726 .vs. 2x(512) for 2 2x8. roughly 1.7 times stiffer, that is.

You can "haunch" the ends of the 2x12 to 8 inches - it should be done as a taper, not as a notch - 45 degrees, not 90 degrees - it's actually stronger that way, which is a bit counter-intuitive).

Joist hangers may offer other options to retain full depth to the ends.