Effective communication is fundamental to any interaction, yet challenges arise when participants do not share a common language. Automatic translation systems offer a powerful solution to bridge language barriers in such scenarios, but they introduce errors that can lead to misunderstandings and conversation breakdown. A key issue is that current systems fail to incorporate the rich contextual information necessary to resolve ambiguities and omitted details, resulting in literal, inappropriate, or misaligned translations. In this work, we present a framework to improve large language model-based translation systems by incorporating contextual information in bilingual conversational settings. During training, we leverage context-augmented parallel data, which allows the model to generate translations sensitive to conversational history. During inference, we perform quality-aware decoding with context-aware metrics to select the optimal translation from a pool of candidates. We validate both components of our framework on two task-oriented domains: customer chat and user-assistant interaction. Across both settings, our framework consistently results in better translations than state-of-the-art systems like GPT-4o and TowerInstruct, as measured by multiple automatic translation quality metrics on several language pairs. We also show that the resulting model leverages context in an intended and interpretable way, improving consistency between the conveyed message and the generated translations.