At that age (3 years), the issue is not what they'd would like (mostly everything), but what you too would enjoy with them, or at least not be too hassled enabling their enjoyment.
Weather-wise, South Florida is the place to be for all of winter. I used to be in NJ, and cannot describe our bliss the first winter we spent here!
As for things to do, Disney (or Universal or Seaworld) might still have things to enjoy for them, but for that expense, and that crowd, you might want to wait a few years to make it worth the hassle - I suggest 7 or 8 the ideal starting age.
Instead, you could explore visiting places near beaches, so you could spend a lot of time on nice warm sand/water, and also visit. Eg - Sarasota /Naples (west FL) has great (white) beaches, and some nice museums and Zoo. There's this museum called G-Wiz that would keep any kid (and parent) enthralled for most of a day. We have similar museums in Miami and Ft Lauderdale - and they are not boring and musty like some museums I could name; The Discovery & Science museum or the Childrens Museum are all really fun houses. Then there's the Seaquarium in Miami - similar to Sea-world but smaller (and hence, cheaper).
Note - all these activities would involve roaming part of the day in the nice sun, and you will thank me later for that

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Not all of miami beaches are risque, and Ft Lauderdale can be fun any time of the year. The water is warm (unlike the west coast), and there are lots of state parks you could go boating in, and even try out para-sailing.
I don't think you could go wrong just planning to spend time in SoFL, and let things take you where they will. No warm clothes required!