Single-reference techniques based on coupled-cluster (CC) theory, in the forms of linear response (LR) or equation of motion (EOM), are highly accurate and widely used approaches for modeling valence absorption spectra. Unfortunately, these equations with singles and doubles (LR-CCSD and EOM-CCSD) scale as O(N⁶), which may be prohibitively expensive for the study of high-energy excited states using a conventional eigensolver. In this paper, we present an energy-specific non-Hermitian eigensolver that is able to obtain high-energy excited states (e.g., XAS K-edge spectrum) at low computational cost. In addition, we also introduce an improved trial vector for iteratively solving the EOM-CCSD equation with a focus on high-energy eigenstates. The energy-specific EOM-CCSD approach and its low-scaling alternatives are applied to calculations of carbon, nitrogen, oxygen, and sulfur K-edge excitations. The results are compared to other implementations of CCSD for excited states, energy-specific linear response time-dependent density functional theory (TDDFT), and experimental results with multiple statistical metrics are presented and evaluated.