If a friendship makes you feel uncomfortable, pay attention to your instincts.
I feel sorry for the neighbor who is here in this country alone. The writer didn't mention if her children were uncomfortable being showered with love "too soon," but if they weren't, I hope the writer kept her discomfort to herself. The only solution to these different styles of showing affection -- which could be cultural, or a reflection of loneliness, or simply differing personalities and family traditions -- is to meet when the children are in school. Sad, esp because I sense the neighbor was simply expressing affection in a way that her family would.
The first thing I noticed is she moved here from another country. She may be unfamiliar with US customs...or not.