I would take the tabs out. You're right that the TFP philosophy ultimately is to understand what you're putting in your pool and what it's going to do for you. But for now, keep things simple and stick to liquid chlorine. Save the tabs for when you're away on vacation.
There isn't anything algaecide is going to do for you at this point that chlorine won't. TFPers who use algaecide generally only do so when closing, and only a very specific type.
It's hard to say if what's in the sock is algae or pollen (my socks are full of pollen right now). I'm sure your filter was picking up the dead algae before, it's just that new algae was growing so fast it couldn't keep up.
One thing that makes it easier to see improvement in water quality is to take a picture with something like a skimmer net or leaf rake in the pool so that you can see the view of the depths improve.