Recent developments of self-interference suppression techniques have enabled practical implementations of full-duplex (FD) cognitive-radio (CR) communication systems. Such systems can significantly enhance spectrum utilization by allowing a CR user to simultaneously transmit and receive over the same frequency channel. However, the CR FD-capabilities challenge the effectiveness of existing CR-based routing protocols as joint FD-aware channel-assignment and routing are essential to enhance network performance. In this letter, we propose a joint FD-aware channel-assignment and route selection protocol in FD-based CR networks (CRNs) under time-varying channel conditions and transmission rates. Specifically, for a given set of paths between a CR source-destination pair, our protocol computes the channel-assignment over each path that maximizes the end-to-end network throughput subject to interference constraints. This assignment problem is shown to be an NP-hard binary linear programming (BLP) that can be sub-optimally solved in polynomial-time using the sequential-fixing procedure. Then, our protocol determines the path with the highest end-to-end network throughput. Simulation results reveal that our proposed routing protocol significantly improves the end-to-end throughput compared to previous FD-aware routing protocols.