We study energy management policies for the compression and transmission of source data collected by an energy-harvesting sensor node with a finite energy buffer (e.g., rechargeable battery) and a finite data buffer (memory) between source encoder and channel encoder. The sensor node can adapt the source and channel coding rates depending on the observation and channel states. In such a system, the absence of precise information about the amount of energy available in the future is a key challenge. We provide analytical bounds and scaling laws for the average distortion that depend on the size of the energy and data buffers. We furthermore design a resource allocation policy that achieves almost optimal distortion scaling. Our results demonstrate that the energy leakage of state of art energy management policies can be avoided by jointly controlling the source and channel coding rates.