We analyze the properties of particles trapped in three-dimensional potentials formed from superimposed Gaussian beams, fully taking into account effects of potential anharmonicity and nonseparability. Although these effects are negligible in more conventional optical lattice experiments, they are essential for emerging ultracold-atom developments. We focus in particular on two potentials utilized in current ultracold-atom experiments: arrays of tightly focused optical tweezers and a one-dimensional optical lattice with transverse Gaussian confinement and highly excited transverse modes. Our main numerical tools are discrete variable representations (DVRs), which combine many favorable features of spectral and grid-based methods, such as the computational advantage of exponential convergence and the convenience of an analytical representation of Hamiltonian matrix elements. Optimizations, such as symmetry adaptations and variational methods built on top of DVR methods, are presented and their convergence properties discussed. We also present a quantitative analysis of the degree of nonseparability of eigenstates, borrowing ideas from the theory of matrix product states, leading to both conceptual and computational gains. Beyond developing numerical methodologies, we present results for construction of optimally localized Wannier functions and tunneling and interaction matrix elements in optical lattices and tweezers relevant for constructing effective models for many-body physics.