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Natural language (NL) explanations of model predictions are gaining popularity as a means to understand and verify decisions made by large black-box pre-trained models, for NLP tasks such as Question Answering (QA) and Fact Verification. Recently, pre-trained sequence to sequence (seq2seq) models have proven to be very effective in jointly making predictions, as well as generating NL explanations. However, these models have many shortcomings; they can fabricate explanations even for incorrect predictions, they are difficult to adapt to long input documents, and their training requires a large amount of labeled data. In this paper, we develop FiD-Ex, which addresses these shortcomings for seq2seq models by: 1) introducing sentence markers to eliminate explanation fabrication by encouraging extractive generation, 2) using the fusion-in-decoder architecture to handle long input contexts, and 3) intermediate fine-tuning on re-structured open domain QA datasets to improve few-shot performance. FiD-Ex significantly improves over prior work in terms of explanation metrics and task accuracy, on multiple tasks from the ERASER explainability benchmark, both in the fully supervised and in the few-shot settings.