The interplay of rotating and counterrotating interactions with two-photon relaxation is a central problem in quantum optics and quantum information processing. This work studies these processes using the anisotropic Rabi model, analyzes excitation-relaxation dynamics through the Lindblad master equation, and derives a non-Hermitian effective Hamiltonian for physical insight. The competing processes show parity-sensitive transient and steady-state behavior, including dynamical patterns reminiscent of quantum phase transitions.