Man, well said. People who "over engage" are doing it out of a sense of kindness, but you're right that it feels hollow and is really just about them.
I think the solution to this is basically what you're doing. Build small connections via whatever engagement mechanism you can and let them organically grow into meaningful ones. Jumping from zero to pretending you have a meaningful connection is exactly why those gestures feel hollow. There is no shortcut, it takes time.
Sounds like you're making those initial connections with your signs, which I think is great.