We present a study of excited charm-strange baryon states produced in e+e−e+e− annihilations at or near a center-of-mass energy of 10.58GeV10.58GeV, in a data sample with an integrated luminosity of 384fb−1384fb−1 recorded with the BABAR detector at the PEP-II e+e−e+e− storage rings at the Stanford Linear Accelerator Center. We study strong decays of charm-strange baryons to Λ+cK0SΛc+KS0, Λ+cK−Λc+K−, Λ+cK−π+Λc+K−π+, Λ+cK0Sπ−Λc+KS0π−, Λ+cK0Sπ−π+Λc+KS0π−π+, and Λ+cK−π+π−Λc+K−π+π−. This study confirms the existence of the states Ξc(2980)+Ξc(2980)+, Ξc(3077)+Ξc(3077)+, and Ξc(3077)0Ξc(3077)0, with a more accurate determination of the Ξc(2980)+Ξc(2980)+ mass and width. We also present evidence for two new states, Ξc(3055)+Ξc(3055)+ and Ξc(3123)+Ξc(3123)+, decaying through the intermediate resonant modes Σc(2455)++K−Σc(2455)++K− and Σc(2520)++K−Σc(2520)++K−, respectively. For each of these baryons, we measure the yield in each final state, determine the statistical significance, and calculate the product of the production cross-section and branching fractions. We also measure the masses and widths of these excited charm-strange baryons.