Florida is hot , but it’s not THAT hot. And the humidity is bearable.
First things first. I think you should address the core problem first , hydration and then the secondary matters: exercise gear , protection and timing of training.
Hydration:
Prehydrate. This IMO will take at least 6-12 hours to get your electrolyte levels optimal and should be constantly continued so you can train in the heat ANYTIME.
What that means , is ensuring that what you drink contains (preferably ) all of the four electrolytes…sodium , magnesium , potassium and calcium.
That does not mean drinking 6x1L of Gatorade daily ! Diluted 100% ( unsweeetened ) juices are a great addition to your hydration needs. I personally add 5 tablespoons of lemon juice to 1.5Litres of water and drink this throughout the day…and night.
I am presuming you sleep with airconditioning on at night due to the heat ? Because you do , that means that you will be far from optimally hydrated if you sleep 6-8 hours soundly in an airconditioned environment without hydrating.
Hydrate throughout the day. Urine should be clear to pale to indicate that you are not dehydrated.
Water DOES NOT hydrate…it is electrolyte neutral. So drinking 1L upon rising or an hour prior to training will do you little good. Water will “facillitate” hydration if mixed with electrolytes. But ALONE will not hydrate. Research Hyponaetremia and visit some medical tents in hotter climates when you see endurance races going on , that’ll convince you pretty quickly.
Minimize caffeine intake and alcohol as both are diuretics and induce dehydration.
Problem1 :
Because you perspire alot , weigh yourself before and after training as often as possible and record your hydration status before and your weight. Same post exercise and indicate quantity of liquids consumed and what you consumed and urination stops…if any.
You should be able to work out very quickly , your electrolyte requirements.
Regarding timing of training… personally I think that ±80% of training can be done at cooler times , but unless races take place at those times ( as you have indicated ) acclimatizing is necessary.
I find that provided you have prepared correctly for the hotter runs , you should be able to get through them.
I would recommend wearing sunblock , a breathable light coloured hat , UV protected sunglasses , a breathable singlet ( or not ) , proper running shorts AND a small waterproof coin pouch ( or ziplock bag that you can minimize in size ) to carry $5 so that you can run and stop by gas stations or whatever is CONVENIENT every 30-45 mins at first to buy water and pop a couple salt tabs or a NUUN which you will have carried with you in your pouch.
Perhaps start at 30-45 min runs and increase as you feel more acclimatized.
Prehydration , correctly hydrating whilst training and post-exercise hydration ,will go a long way towards coping in extreme heat.
Note: I am not a sports doctor , exercise physician , physiotherapist etc.
Just a 43 year old AG athlete who lives ,trains ad races in a very hot climate for the past 16 years and has learnt the hard way 
Good luck
Terry
