The recently developed and implemented orthogonally spin-adapted state-universal (SU) coupled-cluster (CC) theory using a model space spanned by two closed-shell configurations and involving singly and doubly excited clusters (SU CCSD) is applied to calculate static properties of a few typical quasidegenerate systems, for which the range of quasidegeneracy can be continuously varied by changing their geometries. Electrostatic multipole moments and polarizabilities are calculated for the two lowest totally symmetric singlet states of the so-called H4 model consisting of two interacting hydrogen molecules in various geometrical arrangements and for methylene at equilibrium geometry. In both cases, the double-zeta plus polarization basis set is employed and the properties are evaluated by the finite-field method. We discuss the role of orbital relaxation in the SU CC property calculations and compare our results with available single-reference (SR) CC, many-body perturbation theory, and configuration interaction (CI) data, including the full CI results providing the exact solution for the given models. The studied systems enable us to examine several important aspects that are encountered in property calculations when using the SU CC approach. In particular, the strongly degenerate region of the H4 model provides us with several physically interesting situations, involving broken-symmetry solutions and a wrong sign or a wrong order of magnitude of the multipole moments at the Hartree-Fock (HF) or even SR CC level of approximation. Our results indicate that SU CCSD provides accurate values for various electrostatic properties in both degenerate and nondegenerate regimes, regardless of whether the relaxed or nonrelaxed orbitals are employed. At the same time, it gives very good property values for excited states. Finally, even when HF or SR CCSD results are qualitatively wrong due to the symmetry breaking, SU CCSD is capable of correcting this behavior.