  Aims: The analysis of combined photometry and spectroscopy of eccentric
eclipsing binary systems facilitates the derivation of precise values for the
parameters of the component stars and their orbits, thereby providing stringent
tests of theories of stellar structure and evolution. In this paper two
eccentric eclipsing binary systems, TYC 5378-1590-1 and TYC 8378-252-1, are
studied in detail for the first time. Methods: Radial velocities were obtained
using cross-correlation methods applied to mid-resolution spectra covering
almost the entire orbital phase domains of these two systems. TESS photometry
was used for the analysis of TYC 5378-1590-1, whereas ASAS-SN photometry was
used for the analysis of TYC 8378-252-1. Results: We obtained the first precise
derivation of the physical parameters of these systems. Both systems display
moderately eccentric orbits (e = 0.3 and 0.2) with periods of 3.7323 and 2.8776
days, respectively. The apsidal motion is very slow, with a duration of several
centuries. We present two models for the apsidal motion of TYC 5378-1590-1. The
internal structure constant derived from observations for TYC 8378-252-1 is
approximately 11% lower than theoretical predictions. We discuss possible
reasons for this discrepancy. Our analysis indicates that the components of
both systems are on the main sequence. The components of TYC 5378-1590-1 are
relatively young stars (age 600 Myr) close to the ZAMS, whereas the components
of TYC 8378-252-1 are relatively old stars (age 4 Gyr) close to the TAMS. Our
finding that the circularization timescale for TYC 5378-1590-1 is 200 times
longer than its evolutionary age is compatible with theory; however, our the
evolutionary age of TYC 8378-252-1 is approximately ten times longer than the
circulation age, while its orbital eccentricity is quite high (e= 0.2),
challenges the present theories of circularization.
