In recent years, pretrained language models have revolutionized the NLP world, while achieving state-of-the-art performance in various downstream tasks. However, in many cases, these models do not perform well when labeled data is scarce and the model is expected to perform in the zero or few shot setting. Recently, several works have shown that continual pretraining or performing a second phase of pretraining (inter-training), which is better aligned with the downstream task, can lead to improved results, especially in the scarce data setting. Here, we propose to leverage sentiment-carrying discoursemarkers to generate large-scale weakly-labeled data, which in turn can be used to adapt general-purpose language models to the task of sentiment classification. In addition, we propose a new method for adapting sentiment classification models to new domains. This method is based on automatic identification of domain-specific sentiment-carrying discourse markers. Extensive experimental results show the value of our approach on various benchmark datasets. Code, models and data are available at https://github.com/ibm/tslm-discourse-markers.