Oooh, pick me, pick me! (for travel advice, that is...)
Yes, Atlantis is fantastic about allergies, as is Disney's Vero Beach Resort and a select group of cruise ships. The Fairmont Southampton in Bermuda is also quite good with allergies. There are a few all-inclusive resorts I could recommend, as well -- though that list is extremely limited not always by the resort's capabilities but by the standard of medical care nearby.
The Disney ships are truly the best with school-age kids (age 3 to 17) and food allergies (though I don't know how old your kids are...) and their medical facilities are also the best equipped for kids. And yes, they can handle anaphylaxis and have full life support and med-evac capabilities with the US or local Coast Guard when necessary. You could pick a cruise that docks in port most days, so you know that hospitals are easily accessible -- though in some ports they would actually transport you back to the ship because its medical facilities are better than some local hospitals (in smaller, more remote areas, that is.) A Transatlantic cruise may not be a good idea for your first cruise, in that regard, but many Caribbean or European itineraries would be nice.
There's also a restaurant chain in several beach communities that is excellent with food allergies: Lulu's:
https://lulubuffett.com Clearwater, Florida is another nice option, with plenty to do -- though you'd probably want to rent a villa and cook your own food, so that may not be ideal unless you split the stay at Disney or a cruise (a 2-hour drive away), where the chefs will cater to everyone.
Hmmm... so many options... but you'll want to book now as Spring Break is prime vacation time and many places book solid up to a year in advance - seriously. I just looked up one popular destination for Spring Break dates here and they only have large suites left, at a ridiculously high rate.