We present our analysis of VLT/UVES and X-shooter observations of six very metal-poor stars, including four stars at [Fe/H] ≈ -3 in the Fornax and Carina dwarf spheroidal (dSph) galaxies. To date, this metallicity range in these two galaxies has not yet been investigated fully, or at all in some cases. The chemical abundances of 25 elements are presented, based on 1D and local thermodynamic equilibrium (LTE) model atmospheres. We discuss the different elemental groups, and find that α- and iron-peak elements in these two systems are generally in good agreement with the Milky Way halo at the same metallicity. Our analysis reveals that none of the six stars we studied exhibits carbon enhancement, which is noteworthy given the prevalence of carbon-enhanced metal-poor stars without s-process enhancement (CEMP-no) in the Galaxy at similarly low metallicities. Our compilation of literature data shows that the fraction of CEMP-no stars in dSph galaxies is significantly lower than in the Milky Way, and than in ultra-faint dwarf galaxies. Furthermore, we report the discovery of the lowest metallicity, [Fe/H] = -2.92, r-process rich (r-I) star in a dSph galaxy. This star, fnx-06-019, has [Eu/Fe] = +0.8, and also shows enhancement of La, Nd, and Dy, [X/Fe] > +0.5. Our new data in Carina and Fornax help populate the extremely low metallicity range in dSph galaxies, and add to the evidence of a low fraction of CEMP-no stars in these systems.