Deep operator networks (DeepONets, DONs) offer a distinct advantage over traditional neural networks in their ability to be trained on multi-resolution data. This property becomes especially relevant in real-world scenarios where high-resolution measurements are difficult to obtain, while low-resolution data is more readily available. Nevertheless, DeepONets alone often struggle to capture and maintain dependencies over long sequences compared to other state-of-the-art algorithms. We propose a novel framework that leverages multi-resolution data in training and provides precise models when dealing with limited high-resolution data. We achieve this through extending the DeepONet architecture with a long short-term memory network (LSTM), coined DON-LSTM, and training it in a three-step procedure that utilizes data of different levels of granularity. Combining these two architectures, we equip the network with explicit mechanisms to leverage multi-resolution data, as well as capture temporal dependencies in long sequences. We test our method on long-time-evolution modeling of multiple non-linear partial differential equations and show that the proposed multi-resolution DON-LSTM achieves significantly lower generalization error and requires fewer high-resolution samples compared to its vanilla counterparts.