Both should go on before the Tru-Oil. You can scrape and sand the old Tru-Oil off, dye the wood add the skins and reapply the Tru-Oil.
After you put the dye on and it is completely dry, add a thin coat of TBIII over the back and let it cure completely. This will act as a vapor barrier for when you glue the skins on. If you don't do this first the moisture from the glue and skins will be sucked into the dry bow wood.