A routing scheme for wireless sensor networks with mobile sensors and mobile multiple sinks is proposed and studied. The scheme is based on expanding ring search, anycast messaging and reactive mode with maintaining route state information in sensors. As a result of a successful routing request issued by the sensor, it becomes a member of a routing tree with some sink as a root. Anycast messaging is used only at the stage of establishing a path from a sensor to a sink. Replies from sinks are always forwarded in unicast mode. This considerably reduces network traffic and, as a result, energy consumption by sensors. To take into account routing conditions for network nodes in receiving messages from different directions, the receiving area of each node is assumed to consist of a number of sectors, considered as independent links with random change of link states in time. The proposed routing scheme was investigated with the use of a detailed simulation model, implemented in terms of a class of extended Petri nets. In simulation the following performance metrics were investigated versus time-to-live value: response ratio, relative network traffic and relative energy consumption. These metrics were considered for a number of combinations of parameters, such as the number of sinks, sensor availability and link availability. The results of simulation were compared with published characteristics of a similar model, in which sensors do not maintain any routing state information, and is proved to outperform it.