Lee dwarfs everybody, and while it's probably intimidating for them, it's also intimidating for her; they usually have to pause to get their bearings, they often stare. She hates being the most identifiable person in a group and she hates how she has to talk down to them, literally talk down. It makes her feel vulnerable, singled out.
Attracting this much attention, even if it's from people who are only curious, makes it worse.
When she mumbles, "Hi," and awkwardly shakes Peggy's hand, she has that deer-in-headlights look on her face she typically does when she's been thrust into a social situation she is not prepared for. Probably that is not a reaction Peggy is used to receiving. There's an awkward silence afterwards — Lee does not know what to say to these people, these strangers. She knows The Rules for interacting with normal people, it's a script she's memorised but can never quite get herself to recite because she feels like an asshole.