No they do, thats one of the key innovations of LLMs the feed forward step where they propagate information from related words into each other based on context. from https://www.understandingai.org/p/large-language-models-explained-with?r=cfv1p

For example, in the previous section we showed a hypothetical transformer figuring out that in the partial sentence “John wants his bank to cash the,” his refers to John. Here’s what that might look like under the hood. The query vector for his might effectively say “I’m seeking: a noun describing a male person.” The key vector for John might effectively say “I am: a noun describing a male person.” The network would detect that these two vectors match and move information about the vector for John into the vector for his.