Cell-free (CF) massive multiple-input-multiple-output (mMIMO) deployments are usually investigated with half-duplex nodes and high-capacity fronthaul links. To leverage the possible gains in throughput and energy efficiency (EE) of full-duplex (FD) communications, we consider a FD CF mMIMO system with practical limited-capacity fronthaul links. We derive closed-form spectral efficiency (SE) lower bounds for this system with maximum-ratio combining/maximum-ratio transmission processing and optimal uniform quantization. We then optimize the weighted sum EE (WSEE) via downlink and uplink power control by using a two-layered approach: the first layer formulates the optimization as a generalized convex program, while the {second layer} solves the optimization decentrally using the alternating direction method of multipliers. We analytically show that the proposed two-layered formulation yields a Karush-Kuhn-Tucker point of the original WSEE optimization. We numerically show the influence of weights on the individual EE of the users, which demonstrates the utility of the WSEE metric to incorporate heterogeneous EE requirements of users. We show that low fronthaul capacity reduces the number of users each AP can support, and the cell-free system, consequently, becomes user-centric.