Autonomous fabric manipulation is a challenging task due to complex dynamics and potential self-occlusion during fabric handling. An intuitive method of fabric-folding manipulation first involves obtaining a smooth and unfolded fabric configuration before the folding process begins. However, the combination of quasi-static actions like pick & place and dynamic action like fling proves inadequate in effectively unfolding long-sleeved T-shirts with sleeves mostly tucked inside the garment. To address this limitation, this paper introduces an enhanced quasi-static action called pick & drag, specifically designed to handle this type of fabric configuration. Additionally, an efficient dual-arm manipulation system is designed in this paper, which combines quasi-static (including pick & place and pick & drag) and dynamic fling actions to flexibly manipulate fabrics into unfolded and smooth configurations. Subsequently, once it is confirmed that the fabric is sufficiently unfolded and all fabric keypoints are detected, the keypoint-based heuristic folding algorithm is employed for the fabric-folding process. To address the scarcity of publicly available keypoint detection datasets for real fabric, we gathered images of various fabric configurations and types in real scenes to create a comprehensive keypoint dataset for fabric folding. This dataset aims to enhance the success rate of keypoint detection. Moreover, we evaluate the effectiveness of our proposed system in real-world settings, where it consistently and reliably unfolds and folds various types of fabrics, including challenging situations such as long-sleeved T-shirts with most parts of sleeves tucked inside the garment. Specifically, our method achieves a coverage rate of 0.822 and a success rate of 0.88 for long-sleeved T-shirts folding. Supplemental materials and dataset are available on our project webpage at https://sites.google.com/view/fabricfolding.