I like the idea that organic ships exist but they are a precursors tech. 

Some types might be only STL and not FTL.

I think the Sathar should have it or some sort of cyborg hybrid. Sathar Brains running the show maybe on their ships.

The Sathar ships have always struck me as more organic than the other races... I like the idea of them being sort of big bio-mechanical cyborg metal worms in space... partially grown, partially outfitted, maybe intelligent, or at least a little xenophobic (sort of the way Talon was designed to be on FarScape by the Peacekeepers). I think the Sathar ships might be an attempt to recover or use precursor tech. 

I also love the old War of the World's ships and I am thinking of making them Living Ships of the Tetrarchs or some other race. 

Big Giant Space Critters could become part of a STL space faring races propulsion system, some examples of this are in sci-fi... where the critter is used to travel great distances.

Vorlon Ships from Bab5, Leviathan from FarScape are 2 good examples, Lexx from Lexx... I think ST even had an example. 

FarScapes: Leviathan ship graveyard: a sacred place where living leviathans bring debree of their dead to for "burial".

There should be limits to what a bio-energy generated jump can do, how often, how far and so on.

How the living ship is powered? How does it jump if it can?

Basically just kicking the idea around...