There are more than 300 documented signed languages worldwide, which are indispensable avenues for computational linguists to study cross-cultural and cross-linguistic factors that affect automatic sign understanding and generation. Yet, these are studied under critically low-resource settings, especially when examining multiple signed languages simultaneously. In this work, we hypothesize that a linguistically informed alignment algorithm can improve the results of sign-to-sign translation models. To this end, we first conduct a qualitative analysis of similarities and differences across three signed languages: American Sign Language (ASL), Chinese Sign Language (CSL), and German Sign Language (DGS). We then introduce a novel generation and alignment algorithm for translating one sign language to another, exploring Large Language Models (LLMs) as intermediary translators and paraphrasers. We also compile a dataset of sign-to-sign translation pairs between these signed languages. Our model trained on this dataset performs well on automatic metrics for sign-to-sign translation and generation. Our code and data will be available for the camera-ready version of the paper.