OK, I’ve only known 3 native Floridians, but based on news I’ve seen from there, it seems most Floridians don’t seem to have a discernable accent (if that’s the correct term). To me, they sound like Californians. Yet, the rest of South, be it Texans or Georgians, I can pick up that they’re from the South. So, why not Florida?
In the panhandle, they tend to have strong Southern accents.
What sort of Floridian? A native from the rural areas sounds very distinctive. It’s classic american Southern, only moreso.
If by “native” you means somebody whose parents noved to Florida from out of state shortly before or after their birth and who grew up in one of the big suburban areaa with all the other transplants, well it’s not too surprising they sound like they’re from nowhere particular.
Everyone has an accent. You’re asking why they don’t have a specific accent, or an accent from a specific family, and that’s because people from southern Florida (that is, below the panhandle) are culturally distinct from the Deep South. They aren’t “away down south in Dixie”, they’re “away down south of Dixie.” 
This is due to southern Florida’s history as a tourist magnet. Everyone from Cubans to old New Yorkers has moved there and displaced the kind of accents you might expect a region that close to Jawjuh to have.
I was born in Florida, grew up in Florida, live in Florida. I do not have a typical “southern drawl.” I will use “ya’ll” on occasion.
Florida is not really part of the Deep South, despite their being part of the Confederacy. We’re more of a melting pot.
Also, we do NOT sound like Californians.
I’m English and I can totally assure you that my speech is totally without accent .
Unlike everyone else.
Same here, I think I sound pretty close to the standard television news anchor accent (what i consider not accented.)
Central Floridians tend to not have much of an accent, but there are plenty parts of Florida that sound much more Southern. Generally the farther away from a big city the more of an accent there will be.
Of course there are plenty of hispanic accents, haitian accents, etc. as well.
The further south you go in Florida, the further north you get…
When I lived in Orlando, it seemed like the accents of white residents were strongly linked to class and occupation. Working-class and lower-middle class whites, and middle-class whites in blue-collar fields (usually construction and the trades) usually spoke in a Georgia-like southern drawl, while white-collar middle-class and upper-income whites tended to speak with a generic “Midwestern” accent.
In some parts of town – the inner eastern, northern and southern suburbs – Southern/Georgia accents were uncommon. In other areas – Apopka, Ocoee, Winter Garden, Clermont, Sanford and trailer park colonies like Bithlo – Southern/Georgia accents were far more common.
Sanford doesn’t really fit into your list, because half of it (Heathrow and the insurance district) is very well-to-do and not-accented.
It’s really not that complicated, though; as people said above, we’ve got lots of newish residents, who tend to sound like people do wherever they came from… except in the Panhandle, which is unpleasant and doesn’t attract new residents. 
Along the lines of what Derleth said…
North Florida was settled by Southerners.
South Florida was settled by Northerners.
It’s that simple.
I’ve made and previously referenced a handy map of the cultures of Florida. We can argue around the borders, but IMO it’s pretty close.
Yeah, but if you get far enough to the south it flips again. Loog at dem peligans fly!
I googled ‘Cracker Country’, but only found a reference to a museum, which didn’t explain much about why it has so many counties marked black on the map. Explanation pleasey? 
Not that simple any more: You’ve got to make space for Cuba Del Norte. And you have to make space for French-Canadians, although only because its against the law to keep 'em out.
It’s the bit that’s full of crackers.
Except it’s not derogatory when you mean Florida Cracker.
Like others have said, Florida is a huge melting pot of people from everywhere. You may find a handful of second genereation Floridians if you look and even less third generation. Major cities like Orlando, Tampa, Miami got huge not because the locals had more kids but because people moved in from elsewhere.
When I lived and worked throughout the state in the late 90s I hired a lot of “kids” (17-20 year-olds) who were born and raised there but anyone in their mid 20s and older came from somewhere else.
Prepare to be horrified, then, because I can’t distinguish between you and Canadians!