Unmanned and intelligent agricultural systems are crucial for enhancing agricultural efficiency and for helping mitigate the effect of labor shortage. However, unlike urban environments, agricultural fields impose distinct and unique challenges on autonomous robotic systems, such as the unstructured and dynamic nature of the environment, the rough and uneven terrain, and the resulting non-smooth robot motion. To address these challenges, this work introduces an adaptive LiDAR odometry and mapping framework tailored for autonomous agricultural mobile robots operating in complex agricultural environments. The proposed framework consists of a robust LiDAR odometry algorithm based on dense Generalized-ICP scan matching, and an adaptive mapping module that considers motion stability and point cloud consistency for selective map updates. The key design principle of this framework is to prioritize the incremental consistency of the map by rejecting motion-distorted points and sparse dynamic objects, which in turn leads to high accuracy in odometry estimated from scan matching against the map. The effectiveness of the proposed method is validated via extensive evaluation against state-of-the-art methods on field datasets collected in real-world agricultural environments featuring various planting types, terrain types, and robot motion profiles. Results demonstrate that our method can achieve accurate odometry estimation and mapping results consistently and robustly across diverse agricultural settings, whereas other methods are sensitive to abrupt robot motion and accumulated drift in unstructured environments. Further, the computational efficiency of our method is competitive compared with other methods. The source code of the developed method and the associated field dataset are publicly available at https://github.com/UCR-Robotics/AG-LOAM.