Personally I suspect that the organic ships would have had some sort of self-healing ability.
That's what organic ships traditionally get anyway.
They do, indirectly. When you pick organic technology you get organic armor and weapons too; the armor is self-healing. At a quick rate too, I noticed.
The Organic Hulls in the Balance Mod add regeneration capacity to organic armor to help to heal just a wee bit faster. They also have a small reduction to maintenence and cost more in organics. Crystalline Hulls reduce the damage from energy weapons slight, cost more in rads and have the same maintenence reduction.