In recent days, the escalating need to seamlessly transfer data traffic without discontinuities across the Internet network has exerted immense pressure on the capacity of these networks. Consequently, this surge in demand has resulted in the disruption of traffic flow continuity. Despite the emergence of intelligent networking technologies such as software-defined networking, network cloudification, and network function virtualization, they still need to improve their performance. Our proposal provides a novel solution to tackle traffic flow continuity by controlling the selected packet header bits (Differentiated Services Code Point (DSCP)) that govern the traffic flow priority. By setting the DSCP bits, we can determine the appropriate precedence level for the data traffic flow. The control and modification of the DSCP bits directly impact the priority assigned to data packets, thereby shaping the traffic flow continuity accordingly. The evaluation and performance analysis of the proposed network were conducted using the Mininet simulator and the MATLAB platform. The outcomes of the comprehensive testing demonstrate that the implementation of our novel priority management technique has successfully reduced queuing delay and minimized packet loss. As a result, the overall data traffic continuity has been significantly enhanced. The obtained results demonstrate that the traffic flow continuity, handled by the Software-Defined Networking (SDN) controller with the support of DSCP modification, has increased by approximately 65% when implementing the proposed priority management based on DSCP bits modification in the SDN network.