Nonlinear, numerical computation with the NIMROD code is used to explore magnetic self-organization during the formation, sustainment, and relaxation of a spheromak with coaxial helicity injection. The formation process is largely insensitive to the local current distribution in the annular current column and details of the injector current trace, instead depending on the peak injected current (specifically λinj/λeig) and the bias flux Ψbias. The injected flux Φinj needed to trigger the column mode instability, the resulting local poloidal flux amplification AΨ, the amount of injected energy retained in the plasma as magnetic energy ηM, and the magnetic helicity per unit of injected energy ηK all scale with λinj/λeig. Both Φinj and ηK scale with the bias flux Ψbias, and ηM and ηK scale inversely with the pre-fill density. Sustained relaxation occurs when the injector current is maintained between two threshold values, 0.8 < λinj/λeig < 1 for the geometry studied. Overall, our results suggest that the performance of the spheromak configuration is very robust and therefore allows for flexibility in the design and operation of a Proof-of-Principle spheromak concept.