In this work, we propose a weak supervision pipeline SWEET: Supervise Weakly for Entity Extraction to fight Trafficking for extracting person names from noisy escort advertisements. Our method combines the simplicity of rule-matching (through antirules, i.e., negated rules) and the generalizability of large language models fine-tuned on benchmark, domain-specific and synthetic datasets, treating them as weak labels. One of the major challenges in this domain is limited labeled data. SWEET addresses this by obtaining multiple weak labels through labeling functions and effectively aggregating them. SWEET outperforms the previous supervised SOTA method for this task by 9% F1 score on domain data and better generalizes to common benchmark datasets. Furthermore, we also release HTGEN, a synthetically generated dataset of escort advertisements (built using ChatGPT) to facilitate further research within the community.