Yes, we do. Or at least I know that I don't. I can reason based on facts I have learned, creating novel insights that nobody taught me previously. LLMs can't do that and have been shown time and again to be completely incapable of reasoning. Therefore, if you can reason, you know you don't work like an LLM.
The idea that humans could "work like" LLMs (or vice versa) is very vague and can be stretched to say pretty much anything. It's a pointless question IMO. I don't think I do, but maybe I really do on the inside and my consciousness makes me think I don't! We don't know.