Intelligent telemedicine technology has been widely applied due to the quick development of the Internet of Things (IoT). The edge-computing scheme can be regarded as a feasible solution to reduce energy consumption and enhance the computing capabilities for the Wireless Body Area Network (WBAN). For an edge-computing-assisted intelligent telemedicine system, a two-layer network architecture composed of WBAN and Edge-Computing Network (ECN) was considered in this paper. Moreover, the age of information (AoI) was adopted to describe the time cost for the TDMA transmission mechanism in WBAN. According to the theoretical analysis, the strategy for resource allocation and data offloading in edge-computing-assisted intelligent telemedicine systems can be expressed as a system utility function optimizing problem. To maximize the system utility, an incentive mechanism based on contract theory (CT) was considered to motivate edge servers (ESs) to participate in system cooperation. To minimize the system cost, a cooperative game was developed to address the slot allocation in WBAN, while a bilateral matching game was utilized to optimize the data offloading problem in ECN. Simulation results have verified the effectiveness of the strategy proposed in terms of the system utility.