Text Simplification improves the readability of sentences through several rewriting transformations, such as lexical paraphrasing, deletion, and splitting. Current simplification systems are predominantly sequence-to-sequence models that are trained end-to-end to perform all these operations simultaneously. However, such systems limit themselves to mostly deleting words and cannot easily adapt to the requirements of different target audiences. In this paper, we propose a novel hybrid approach that leverages linguistically-motivated rules for splitting and deletion, and couples them with a neural paraphrasing model to produce varied rewriting styles. We introduce a new data augmentation method to improve the paraphrasing capability of our model. Through automatic and manual evaluations, we show that our proposed model establishes a new state-of-the art for the task, paraphrasing more often than the existing systems, and can control the degree of each simplification operation applied to the input texts.