Sign language understanding has made significant strides; however, there is still no viable solution for generating sign sequences directly from entire spoken content, e.g., text or speech. In this paper, we propose a unified framework for continuous sign language production, easing communication between sign and non-sign language users. In particular, a sequence diffusion model, utilizing embeddings extracted from text or speech, is crafted to generate sign predictions step by step. Moreover, by creating a joint embedding space for text, audio, and sign, we bind these modalities and leverage the semantic consistency among them to provide informative feedback for the model training. This embedding-consistency learning strategy minimizes the reliance on sign triplets and ensures continuous model refinement, even with a missing audio modality. Experiments on How2Sign and PHOENIX14T datasets demonstrate that our model achieves competitive performance in sign language production.