Modifications of the nonlinear Schrödinger (MNLS) model tψ(x,t) + a x2ψ(x,t)-[IV I|ψ|2]ψ(x,t) = 0, where ψ a., and V: IR+ →IR, are considered. We show that the MNLS models possess infinite towers of quasi-conservation laws for soliton-type configurations with a special complex conjugation, shifted parity and delayed time reversion (sd) symmetry. Infinite towers of anomalous charges appear even in the standard NLS model for sd invariant N-bright solitons. The true conserved charges emerge through some kind of anomaly cancellation mechanism. Our analytical results are supported by numerical simulations of two-bright-soliton scatterings with potential V =-2 2+(|ψ|2)2+, IR, > 0. Our numerical simulations show the elastic scattering of bright solitons for a wide range of values of the set {I} and a variety of amplitudes and relative velocities. The MNLS-type systems are quite ubiquitous, and so, our results may find potential applications in several areas of nonlinear physics, such as Bose-Einstein condensation, superconductivity, soliton turbulence and the triality among gauge theories, integrable models and gravity theories.