If it were me I wouldn't move far away into anyones home with my child. I don't care how well I think I know them. Especially in light of your situation. I'm sorry, but I really see no need to rush these things. Being in the same area, even living next door seems far more approriate and safe.
As for the lying. Yes I know, it sucks, but I would suck it up and do it for my kid. I do it for him now, and for my partners. It really goes against my grain, but I see it as a necessary thing for the survival of the life we are loving right now. What they don't know won't hurt us.
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