The recent pandemic along with the rapid increase in the number of patients that require continuous remote monitoring imposes several challenges to support the high quality of services (QoS) in remote health applications. Remote-health (r-health) systems typically demand intense data collection from different locations within a strict time constraint to support sustainable health services. On the contrary, the end-users with mobile devices have limited batteries that need to run for a long time, while continuously acquiring and transmitting health-related information. Thus, this paper proposes an adaptive deep reinforcement learning (DRL) framework for network selection over heteroge-neous r-health systems to enable continuous remote monitoring for patients with chronic diseases. The proposed framework allows for selecting the optimal network(s) that maximizes the accumulative reward of the patients while considering the patients' state. Moreover, it adopts an adaptive compression scheme at the patient level to further optimize the energy consumption, cost, and latency. Our results depict that the proposed framework outperforms the state-of-the-art techniques in terms of battery lifetime and reward maximization. This work was made possible by NPRP grant # NPRP12S-0305-190231 from the Qatar National Research Fund (a member of Qatar Foundation). The findings achieved herein are solely the responsibility of the authors.