Most of the issues come from trying to model the 3d situation with a 1d potential well.
Some notes
Your model is assuming(and enforcing)spherical symmetry, most of the real wave functions do not have spherical symmetry, instead involving spherical harmonic functions to parameterize spherical dependency.
Some of your 1d eigenfunctions are antisymmetric, f(-x)=-f(x). These are totally invalid as potential solutions, where we are trying to pretend x is a radius and we have spherical symmetry, and need to be excluded.
Your lowest state is totally artificial, without your registration/softening at 0 it would have negative infinite energy and be infinitely thin (this couldn't be solved). The value of this is totally a result of softening and resolution.
You may need to reconsider the 1d approach. Or introduce something to account for deviations from spherical symmetry.