check the wires..... either ground or power....
my bet is bad wires... you may not be able to see it, but i bet they are corroded internally
also, how are you checking 12v sitting?
a dvom or multi-meter?
batteries should be sitting @ 12.6 with no surface charge, and should be around 14.6 when running
you said it does run and will stay running once fired up?
what about the ign switch?