The quantum mechanical current density induced in a molecule by an external magnetic field is invariant to translations of the coordinate system. This fundamental symmetry is exploited to formally annihilate the diamagnetic contribution to the current density via the approach of "continuous transformation of the origin of the current density-diamagnetic zero" (CTOCD-DZ). The relationships obtained by this method for the magnetic shielding at the nuclei are intrinsically independent of the origin of the coordinate system for any approximate computational scheme relying on the algebraic approximation. The authors report for the first time an extended series of origin-independent estimates of nuclear magnetic shielding constants using the CTOCD-DZ approach at the level of density functional theory (DFT) with four different types of functionals and unrelaxed coupled cluster singles and doubles linear response (CCSD-LR) theory. The results obtained indicate that in the case of DFT the procedure employed is competitive with currently adopted computational methods allowing for basis sets of gauge-including atomic orbitals, whereas larger differences between CTOCD-DZ and common origin CCSD-LR results are observed due to the incomplete fulfillment of hypervirial relations in standard CCSD-LR theory. It was found furthermore that the unrelaxed CCSD-LR calculations predict larger correlation corrections for the shielding constants of almost all nonhydrogen atoms in their set of molecules than the usual relaxed energy derivative CCSD calculations. Finally the results confirm the excellent performance of Keal and Tozer's third functional, in particular, for the multiply bonded systems with a lot of electron correlation, but find also that the simple local density functional gives even better results for the few singly bonded molecules in their study where correlation effects are small.