Vehicular ad hoc networks (VANETs) are getting growing interest as they are expected to play crucial role in making safer, smarter, and more efficient transportation networks. Due to unique characteristics such as sparse topology and intermittent connectivity, Delay Tolerant Network (DTN) routing in VANET becomes an inherent choice and is challenging. However, most of the existing DTN protocols do not accurately discover potential neighbors and, hence, appropriate intermediate nodes for packet transmission. Moreover, these protocols cause unnecessary overhead due to excessive beacon messages. To cope with these challenges, this paper presents a novel framework and an Adaptive Geographical DTN Routing (AGDR) for vehicular DTNs. AGDR exploits node position, current direction, speed, and the predicted direction to carefully select an appropriate intermediate node. Direction indicator light is employed to accurately predict the vehicle future direction so that the forwarding node can relay packets to the desired destination. Simulation experiments confirm the performance supremacy of AGDR compared to contemporary schemes in terms of packet delivery ratio, overhead, and end-to-end delay. Simulation results demonstrate that AGDR improves the packet delivery ratio (5-7%), reduces the overhead (1-5%), and decreases the delay (up to 0.02 ms). Therefore, AGDR improves route stability by reducing the frequency of route failures.