Since futuristic cellular core networks are envisioned to accommodate emerging vertical domain applications with diverse QoS requirements, software-defined networking (SDN) technology has played a vital role in improving the scalability and flexibility of network management. This paper presents the design and development of a service-aware transport layer framework in SDN-enabled cellular core networks. It aims to study and accommodate three typical traffic types: enhanced mobile broadband (eMBB), ultra-reliable and low latency communications (URLLC), and massive machine-type communications (mMTC) in optimal path selection. The proposed framework is able to dynamically allocate network resources to corresponding traffic of above types for satisfying the requirements of each identified flow, using a multi-level queue management mechanism to alleviate the flow dropping problem when the network becomes congested. For experiment and verification, a testbed platform was built to evaluate the functionalities of developed modules, the performance of the diffserv-based optimal path selection, and the operation of multi-level queue management. The experiment results demonstrate that the proposed algorithms could obtain a higher accepting ratio than the other ones in different network conditions.