With the advancement of space technology and satellite communications, low-Earth-orbit (LEO) satellite networks have experienced rapid development in the past decade. In the vision of 6G, LEO satellite networks play an important role in future 6G networks. On the other hand, a variety of applications, including many delay-sensitive applications, are continuously emerging. Due to the highly dynamic nature of LEO satellite networks, supporting time-deterministic services in such networks is challenging. However, we can provide latency guarantees for most delay-sensitive applications through data plane traffic shaping and control plane routing optimization. This paper addresses the routing optimization problem for time-sensitive (TS) flows in software-defined low-Earth-orbit (LEO) satellite networks. We model the problem as an integer linear programming (ILP) model aiming to minimize path handovers and maximum link utilization while meeting TS flow latency constraints. Since this problem is NP-hard, we design an efficient longest continuous path (LCP) approximation algorithm. LCP selects the longest valid path in each topology snapshot that satisfies delay constraints. An auxiliary graph then determines the routing sequence with minimized handovers. We implement an LEO satellite network testbed with Open vSwitch (OVS) and an open-network operating system (ONOS) controller to evaluate LCP. The results show that LCP reduces the number of path handovers by up to 31.7% and keeps the maximum link utilization lowest for more than 75% of the time compared to benchmark algorithms. In summary, LCP achieves excellent path handover optimization and load balancing performance under TS flow latency constraints.