This paper describes Unbabel+IT’s submission to the Chat Shared Task held at the Workshop of Machine Translation 2024. The task focuses on translating customer support chats between agents and customers communicating in different languages. We present two strategies for adapting state-of-the-art language models to better utilize contextual information when translating such conversations. Our training strategy involves finetuning the model on chat datasets with context-augmented instructions, resulting in a specialized model, TOWERCHAT. For inference, we propose a novel quality-aware decoding approach that leverages a context-aware metric, CONTEXTCOMET, to select the optimal translation from a pool of candidates. We evaluate our proposed approach on the official shared task datasets for ten language pairs, showing that our submission consistently outperforms baselines on all and competing systems on 8 out of 10 language pairs across multiple automated metrics. Remarkably, TOWERCHAT outperforms our contrastive submission based on the much larger TOWER-V2-70B model while being 10× smaller. These results underscore the importance of context-aware training and inference in handling complex bilingual dialogues.