BTW - the box you're talking about, if it's what I think it is, is the evaporator. The "air handler" is a term used for the entire bundle of equipment which includes the air inlet, the blower/furnace unit, the evaporator and the outlet plenum where all the hoses are attached.
If the evaporator box is sweating, then either there's a massive air leak - which you should be able to feel, or the handler box isn't insulated, which it should have been from the factory. If the unit is newer, there should be an access port you can open to look inside at the airflow inlet side of the evaporator. Look inside to see if the box is insulated. It should be. If it's not, you need to get it insulated, pronto.
If you've got a cold air leak, the chilling of the outside metal will cause condensation, but usually, that's not enough to cause a drip pan to fill - the attic temps will usually evaporate moderate surface condensation before it can accumulate, and in FL you should have ample attic temps.
The overflow pan is a precautionary device - it is not designed to serve as the primary condensate drain from the evaporator coil - WHICH WILL produce GALLONS of condensate on a daily basis, especially in a hot humid environment like FL. It has a level switch that cuts off the outside AC when it gets full, because it should only fill up if the main drain pan (which is integral with the evaporator box) gets a hole, or the drain clogs.
Here in Houston, many houses, mine included have separate drain lines for the drip pan that exit the house in some visible location, usually around the main entrance or a kitchen window. If the homeowner sees water coming out of that pipe, they're supposed to know to call the A/C guy in...
If you're getting condensate into the pan, check the evaporator to see if the drain is clogged - from the inside. If it's not, you've got a "holey" pan, and I think the only solution there is a new evaporator...
EDIT - I just noticed your original post - "on it's side" - evaps should only be installed with the drain pan on the bottom. There are horizontal evaps and vertical evaps. Still, I suspect that you've either got a clogged drain from the evap pan, or the pan has a hole. Access to the pan isn't usually possible without a lot of deconstructive surgery, and then, the current hole is just a precursor to more holes in the future.