First thing you should do is have an arborist look at your tree. It may need to be thinned out.

I think that the shade tolerance of azaleas is over-rated. When I see the ones grown in shade I don't think their foliage compares to those grown with good light, or even full sun. The problem with gardening under a large oak is digging. Usually tree roots are a major impediment. If you're able to clear any out of your way, they'll be back in 3 months. You best bet is some kind of groundcover that spreads vine-like or creeping. By installing very small starts of it, it would spread on its own as long as you choose a plant that likes the light conditions and provide water and fertilizer during the establishment period. It depends on where in Florida you are as to what would work. If you want something low and it grows there, you might consider asiatic jasmine, or even dwarf mondo grass. If you want taller you might consider the ubiquitous sword fern, which you could probably get starts of from the woods or from a neighbor.

If it wouldn't hurt the roots, maybe a large deck area would be nice to create a focal point under the tree...not for the dogs but for anyone who might want to sit out there etc.

For the dirt, I agree maybe creeping groundcovers or some wood chips. Hard not to have dirt, though, w/dogs running around.