Energy use is a crucial design concern in wireless ad hoc networks. The design objectives of energy-aware routing include selecting energy-efficient paths and minimizing the protocol overhead incurred in acquiring such paths. To achieve these goals altogether, we present the design of two energy-aware on-demand routing protocols for different network environments. The key idea behind our design is to adaptively select the subset of nodes required to involve in a route-searching process to acquire a high residual-energy path or the degree to which nodes are required to participate in the process of searching for a low-power path for networks wherein nodes can adaptively adjust their transmission power. Analytical and simulation results are given to demonstrate the high performance of the designed protocols in energy-efficient utilization and in reducing the protocol overhead incurred in acquiring energy-aware routes.