Since you mentioned that French drains aren't for you, another alternative would be rain chains, or maybe a type of dry stream bed type thing directing the water to a holding area. One thing about holding areas in FL is that you're going to have to be really aware of standing water issues such as mosquitos and frogs. Rain barrels would obviously avoid those issues.
That being said, if I'm not mistaken, rain barrels are intended for use with gutters, so if you intend to catch runoff from the roof (which is a LOT of water that would otherwise go to waste) you may end up needing gutters to direct the water into the barrels. There may be a way to use rain chains to direct that water but I am not sure (it's not a great alternative in the part of the country where I live because of our changes of seasons).
Last comment: you really might want to consider regrading those low lying areas near the foundation to prevent foundation damage. Regrading can be as complicated as bringing in heavy equipment or as simple as buying a lot of soil and building up the areas in question and then planting water-loving plants there, along with shrubs that have large root systems.

1) I would reconsider your aversion to gutters/downspouts.

2) If 1 doesn't work, you should grade away from the house, and cover about 1-2 feet with gravel or stones of some sort. At the edge of that stone, you could put something like what's in the link ("Spee-D channel drains" are one name). Basically an in-ground gutter that can get the water to a lower point and away from the house (one hopes). Of course, you need for it to go somewhere, and the street may be acceptable or may not.

Here is a link that might be useful: ground gutters.

You need to address your drainage issues. Rain chains work well for directing the water down...but then it has to go somewhere. Rain barrel, cistern,rain garden, down the hill-- SOMEWHERE. Rain barrels are great, but they fill VERY quickly. Cisterns hold more, but they aren't cheap. You need to figure out your budget...and then start looking at what will fit within that budget.

google the word "Rain Gardens".

Hi
i channel my rain gutters into an above ground garden pool .When they reach the ground they go under releasinfg the water into the pool. Have 1100 gallons of water ,selfcleaning ,garden pool,overflow goes into a marsh /raingarden. kills four birds with one stone lol.
Some problems . A 31 inch rain and i had to collect fish from 5 doors away . Since I keep fish and plants in it can't drain all the water out for irrigation.. And still have found no way to make it rain when I want to lol gary