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Double-lined eclipsing binaries allow the direct determination of masses and radii, which are key to test stellar models. With the launch of the TESS mission, many well-known eclipsing binaries have been observed at higher photometric precision, permitting the improvement of the absolute dimensions determinations. Using TESS data and newly-obtained spectroscopic observations, we aim at determining the masses and radii of the eccentric eclipsing binary systems V889 Aql and V402 Lac, together with their apsidal motion parameters. We modelled simultaneously radial velocity curves and times of eclipse for each target to precisely determine the orbital parameters of the systems, which we used to analyse the light curves and then obtain their absolute dimensions. We compared the obtained values with those predicted by theoretical models. We determined masses and radii of the components of both systems with relative uncertainties lower than 2%. V889 Aql is composed of two stars with masses $2.17\pm0.02$ M$_{\odot}$ and $2.13\pm0.01$ M$_{\odot}$ and radii $1.87\pm0.04$ R$_{\odot}$ and $1.85\pm0.04$ R$_{\odot}$. We found conclusive evidence of the presence of a third body orbiting V889 Aql with a period of 67 years. Based on the detected third light and the absence of signal in the spectra, we suggest that this third body could in turn be a binary composed by two $\sim$1.4 M$_{\odot}$ stars. V402 Lac is composed by two stars with masses $2.80\pm0.05$ M$_{\odot}$ and $2.78\pm0.05$ M$_{\odot}$ and radii $2.38\pm0.03$ R$_{\odot}$ and $2.36\pm0.03$ R$_{\odot}$. The times of minimum light are compatible with the presence of a third body for this system too, although its period is not yet fully sampled. In both cases we have found a good agreement between the observed apsidal motion rates and the model predictions.