Apsidal motion is the precession of the line of apsides in an orbit due to perturbations from general relativistic effects, tidal interactions, rotational effects, and gravitational intervention by other bodies. Observing this phenomenon allows investigation of complex orbital dynamics and the internal structure of stars, and is visualized in an O-C diagram when compared to a calculated linear ephemeris. Historically, apsidal motion has been observed through the analysis of eclipse times to reveal small changes in the interval between primary and secondary eclipses. While successful in characterizing the average value of the two stellar internal structure constants ̄k2,obs, this approach fails to constrain the individual constants k21, k22 of the bodies themselves, and has shown difficulty in binaries with mass ratios q & 1. Here we use the Eclipsing Light Curve program (ELC) to compute individual apsidal constants for a set of eclipsing binary systems. Archival observations of times of minima are collected from the literature for the systems with short periods of apsidal motion EM Car, DR Vul, CW Cep, Y Cyg, and V526 Sgr. New high-cadence photometric data from TESS are obtained from MAST, and eclipse times in the TESS data are determined. Additional times of eclipse for V526 Sgr are collected from the WASP survey. Given the apsidal constants k21 and k22 for each star, along with the orbital elements, the ELC code computes the times of eclipse by integrating the modified Newtonian equations of motion. The optimal values for the apsidal constants and orbital elements are then determined for each system using an optimization code based on a Differential Evolution Monte Carlo Markov Chain technique. Using simulated data sets, we explore the statistical validity of our approach and the limitations of this technique. Provided observed eclipse times are available for most of an apsidal cycle, we show this analysis provides accurate computation of the two individual apsidal constants, and an accurate value of the average of the two constants in agreement with published values. The individual constants do not approach zero, which is an improvement over the previous implementations of this technique.