In the context of the string-inspired running vacuum model (StRVM) of cosmology with anomalies and torsion-induced axion-like fields, we discuss one-loop quantum corrections (either from graviton loops or massive matter quantum fields) to the corresponding energy density. Such corrections are computed within the framework of weak quantum gravity in eras with approximately-de-Sitter-dark-energy dominance, during which the Hubble parameter H(t) varies very slowly with the cosmic time t. The quantum-graviton corrections exhibit a logarithmic dependence on H(t), of the form \(H^n\, \textrm{ln}(H^2)\) , \(n \in 2Z^+\) . In modern eras, the \(n=2\) terns are dominant. It is then demonstrated that such corrections may contribute to the alleviation of both types of the currently observed cosmological tensions, namely \(H_0\) and growth-of-structure tensions. We argue that such an effect is accomplished for a StRVM cosmology based on dynamically-broken primordial (pre-RVM inflation) supergravity (SUGRA). For this model, during the current de-Sitter era, the quantum graviton corrections dominate those due to matter fields, provided the energy scale of the dynamical SUGRA breaking lies a bit lower than the reduced Planck mass, which is a natural assumption, consistent with the transplanckian conjecture.