Ambient backscatter communications has shown the great potentials to achieve spectrum and energy efficient communications for future wireless networks (e.g., 6G). Besides, to fully exploit the ambient RF environment, backscatter communications can be combined with RF-powered cognitive networks which brings more reliable and flexible communications. In such networks, a cognitive user pair could communicate through active transmission or backscattering existing signals. In this work, we investigate the distributed resource allocation in terms of channel selection and backscatter power allocation in such setting. A unified framework integrating stochastic geometry and evolutionary game is proposed to model and analyze the distributed resource allocation. Specifically, stochastic geometry is used to analyze the average performance for users considering different communication modes. Then with the average performance obtained, evolutionary game theory is used to model the competitive resource allocation, and replicator dynamics is used to capture the dynamic change of user selections. Based on the formulated evolutionary game, a distributed channel selection and backscatter power allocation algorithm is proposed. Simulations have been performed which validate the theoretical analysis. Also the effectiveness of the proposed scheme is verified.