This paper presents the current wall-embedded Langmuir probe system installed on the Tokamak à Configuration Variable (TCV), as well as the analysis tool chain used to interpret the current-voltage characteristic obtained when the probes are operated in swept-bias conditions. The analysis is based on a four-parameter fit combined with a minimum temperature approach. In order to reduce the effect of plasma fluctuations and measurement noise, several current-voltage characteristics are usually averaged before proceeding to the fitting. The impact of this procedure on the results is investigated, as well as the possible role of finite resistances in the circuitry, which could lead to an overestimation of the temperature. We study the application of the procedure in a specific regime, the plasma detachment, where results from other diagnostics indicate that the electron temperature derived from the Langmuir probes might be overestimated. To address this issue, we explore other fitting models and, in particular, an extension of the asymmetric double probe fit, which features effects of sheath expansion. We show that these models yield lower temperatures (up to approximately 60%) than the standard analysis in detached conditions, particularly for a temperature peak observed near the plasma strike point, but a discrepancy with other measurements remains. We explore a possible explanation for this observation, the presence of a fast electron population, and assess how robust the different methods are in such conditions.