Opinion formation in online networks involves changes in both beliefs and social ties. Analytical models make it possible to study equilibrium and social cost, but usually represent communication as a fixed numerical update. LLM-driven agents offer a language-based alternative, yet their convergence and collective efficiency remain unclear. We develop the Hybrid Coevolutionary Opinion Game (H-COG), combining cost-minimizing Friedkin-Johnsen agents (Type-C) and Phi-4 language agents (Type-L) in a dynamically rewired K-nearest-neighbor network. We initialize 50 agents with opinions drawn from 5,199 Reddit comments on gun control and abortion. The comments are scored on a continuous [-1,+1] scale using a fine-tuned RoBERTa regressor, and a mixing parameter sets the proportion of each agent type. The experiments cover nine population compositions, three initial network topologies, and two topics. All 540 runs meet the convergence criterion within the simulation horizon. Under Type-L updating, the coevolving network reaches an attractor as reliably as it does under the analytical update rule, making an equilibrium-based efficiency comparison possible. The pooled Price of Anarchy is $5.558 \pm 0.309$ for purely Type-L populations, compared with $1.139 \pm 0.005$ for purely Type-C populations. A decomposition of social cost attributes most of this gap to language agents moving away from their intrinsic opinions, rather than to greater disagreement with their neighbors. The main findings are consistent across the three initial network topologies.

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