It's so interesting to read this, because it shows how vastly different the US is from region to region.
For example, it's incredibly easy to find lamb in the Northeast, or at least where I've lived & traveled in the Northeast. As for supermarkets, I've always thought they were more than plentiful, especially in the suburbs. And there are few national chains the way there are here (Tesco etc) because supermarkets monopolies vary from region to region. I agree with ameriscot that it's worth joining a costco or equivalent. You pay a membership fee and you get a card, and that card entitles you to membership to this ENORMOUS warehouse/supermarket thingie, where you can buy all kinds of things at a discount, including high quality food. If you've never been to a costco, it's quite an experience.
Weirdly I've never noticed any differences between the nappies here & those in the US, and I've never found a leakage problem either. Strange! I used to use pampers & huggies pretty interchangeably.
Never heard of a nanny service in restaurants. People are pretty easy going about taking their kids to restaurants, particularly if it's early and the restaurant is casual.
As for nights out, I have no idea what people do in Florida. I assume what people do is dependent on what's available and local. Where's kateandthegirls? She'd be a hugely valuable source of info, I think.