Ensuring timing guarantees for every individual tasks is critical in real-time systems. Even for periodic tasks, providing timing guarantees for tasks with segmented self-suspending behavior is challenging due to timing anomalies, i.e., the reduction of execution or suspension time of some jobs increases the response time of another job. The release jitter of tasks can add further complexity to the situation, affecting the predictability and timing guarantees of real-time systems. The existing worst-case response time analyses for sporadic self-suspending tasks are only over-approximations and lead to overly pessimistic results. In this work, we address timing anomalies without compromising the worst-case response time (WCRT) analysis when scheduling periodic segmented self-suspending tasks with release jitter. We propose two treatments: segment release time enforcement and segment priority modification, and prove their effectiveness in eliminating timing anomalies. Our evaluation demonstrates that the proposed treatments achieve higher acceptance ratios in terms of schedulability compared to state-of-the-art scheduling algorithms. Additionally, we implement the segment-level fixed-priority scheduling mechanism on RTEMS and verify the validity of our segment priority modification treatment. This work expands our previous conference publication at the 29th IEEE Real-Time and Embedded Technology and Applications Symposium (RTAS 2023), which considers only periodic segmented self-suspending tasks without release jitter.