We present an elemental-abundance analysis, in the near-ultraviolet (NUV) spectral range, for the bright carbon-enhanced metal-poor (CEMP) stars HD 196944 (V=8.40, [Fe/H]=-2.41) and HD 201626 (V=8.16, [Fe/H]=-1.51), based on data acquired with the Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope. Both of these stars belong to the sub-class CEMP-s, and exhibit clear over-abundances of heavy elements associated with production by the slow neutron-capture process. HD 196944 has been well-studied in the optical region, but we add abundance results for six species (Ge, Nb, Mo, Lu, Pt, and Au) that are only accessible in the NUV. In addition, we provide the first determination of its orbital period, P=1325 days. HD 201626 has only a limited number of abundance results based on previous optical work --here we add five new species from the NUV, including Pb. We compare these results with models of binary-system evolution and s-process element production in stars on the asymptotic giant branch, with the goal of explaining their origin and evolution. Our best-fitting models for HD 196944 (M_1,i_=0.9M_{sun}_, M_2,i_=0.86M_{sun}_, for [Fe/H]=-2.2), and HD 201626 (M_1,i_=0.9M_{sun}_, M_2,i_=0.76M_{sun}_ for [Fe/H]=-2.2; M_1,i_=1.6M_{sun}_, M_2,i_=0.59M_{sun}_ for [Fe/H]=-1.5) are consistent with the current accepted scenario for the formation of CEMP-s stars.