Crystal clear water WITH black algae...

A few months ago we discovered what I believed to be was a black algae problem with our pool. We treated it and it SEEMED to be gone, but over time I kept seeing black spots that would not easily brush away. I test my water and keep the chlorine a bit on the high side and the pH balanced...however, still seeing black spots despite shocking, brushing, cleaning the filter, etc.

The other day I tested and chlorine was down to nothing! I am not sure why. Balanced that and the pH and the water was crystal clear. Then saw a LOT of black spots. Shocked the pool...next day crystal clear water but still black spots. So I brushed again (with a wire brush) and double shocked the pool. This morning it is clear again (haven't tested it yet though, only looked outside) but still some black spots (though there don't seem to be as many).

What, if ANYTHING can I do, to get rid of this for GOOD? Everything I read seems to indicate you really CAN'T, you can only control it...if that is the case, if everything is balanced (chemical wise), the pool is crystal clear yet some spots remain, I assume the water is still safe to swim in? Can anyone give me some answers? Thanks!

Edited to add: This is an inground pool, about 15X30. Pool temp ranges from 85-90 (we are in south FL). We have a cover but have left it off for the past few days. I have had the pump running 24/7....