The biases present in training datasets have been shown to be affecting models for a number of tasks such as natural language inference(NLI) and fact verification. While fine-tuning models on additional data has been used to mitigate such biases, a common issue is that of catastrophic forgetting of the original task. In this paper, we show that elastic weight consolidation (EWC) allows fine-tuning of models to mitigate biases for NLI and fact verification while being less susceptible to catastrophic forgetting. In our evaluation on fact verification systems, we show that fine-tuning with EWC Pareto dominates standard fine-tuning, yielding models lower levels of forgetting on the original task for equivalent gains in accuracy on the fine-tuned task. Additionally, we show that systems trained on NLI can be fine-tuned to improve their accuracy on stress test challenge tasks with minimal loss in accuracy on the MultiNLI dataset despite greater domain shift.