Integrating the relaying drone and non-orthogonal multiple access (NOMA) technique into post-disaster emergency communications (PDEComs) is a promising way to accomplish efficient network recovery. Motivated by the above, by optimizing the drone three-dimensional (3D) deployment optimization and spectrum allocation, this paper investigates a quality of service (QoS)-driven sum rate maximization problem for drone-and-NOMA-enhanced PDEComs that aims to improve the data rate of cell edge users (CEUs). Due to the non-deterministic polynomial (NP)-hard characteristics, we first decouple the formulated problem. Next, we obtain the optimal 3D deployment with the aid of a long short-term memory (LSTM)-based recurrent neural network (RNN). Then, we transform the spectrum allocation problem into an optimal matching issue, based on which the Hungarian algorithm is employed to solve it. Finally, the simulation results show that the presented scheme has a significant performance improvement in the sum rate compared with the state-of-the-art works and benchmark scheme. For instance, by adopting the NOMA technique, the sum rate can be increased by 9.72% and the needs of CEUs can be satisfied by enabling the relaying drone. Additionally, the convergence, complexity, and performance gap caused by iterative optimization are discussed and analyzed.