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We investigate the use of orbital-optimized references in conjunction with single-reference coupled-cluster theory with single and double substitutions (CCSD) for the study of core excitations and ionizations of 18 small organic molecules, without any use of response theory or equation-of-motion formalisms. Three schemes are employed to successfully address the convergence difficulties associated with the coupled-cluster equations, and the spin contamination resulting from the use of a spin symmetry-broken reference, in the case of excitations. In order to gauge the inherent potential of the methods studied, an effort is made to provide reasonable basis set limit estimates for the transition energies. Overall, we find that the two best-performing schemes studied here for Delta-CCSD are capable of predicting excitation and ionization energies with errors comparable to experimental accuracies. The proposed Delta-CCSD schemes seem to fare better than the widely used equation-of-motion CCSD (EOM-CCSD) with core-valence separation protocol, with statistical errors being reduced by more than a factor of two when compared to FC-CVS-EOM-CCSD.