Direct Numerical Simulation (DNS) of multiphase flows is hindered by the complex interaction between interfacial tension and fluid dynamics, requiring high resolution of thin, deforming interfaces. This work presents a computationally efficient framework coupling conservative numerical methods with an adaptive, moving deformative mesh strategy. The numerical approach integrates two core components: (i) an algebraic energy-preserving level-set method that ensures mechanical energy conservation via symmetry-preserving schemes, utilizing a fully conservative formulation for convection and surface tension to suppress spurious currents; and (ii) conservative flux-limiting techniques within an algebraic framework, facilitating portable implementation on heterogeneous high-performance computing (HPC) systems. The primary innovation is the integration of these schemes with dynamic mesh adaptation rather than fixed grids. Moving deformative meshes automatically cluster resolution near the interface while maintaining coarser discretization elsewhere, significantly reducing memory footprint and computational overhead. This refinement is driven by an adaptive metric and a variational algorithm that ensures mesh quality and is validated against linear perturbation theory. Validation via canonical benchmarks demonstrates the framework's ability to accurately capture energy transfer between kinetic and surface potential energy over multiple oscillation periods. Results show that the adaptive moving mesh successfully tracks complex interface evolution and maintains high mesh quality without requiring periodic remeshing.