I've been dealing with a cloudy / algae pool recently as well, just got a real test kit 2 days ago and I'm seeing improvement already. Some thoughts from another clueless person that inherited a pool. All of this relates to my system, yours may differ.

On top of your sand filter you should have a gauge that effectively indicates how the system is functioning. When mine is working right it's around 10 psi, if brush and vacuum then it can get up to 30-40psi pretty quick, indicating that the filter is full of crud and needs backwashing. I don't backwash if there's no increase in pressure as I feel this indicates there's still good flow/filtering happening.

There is a prefilter on the pump, with a clear top on it and a filter basket inside. The impeller on the pump can get bound up with junk that gets through the basket, which keeps water from getting sent to the filter correctly. This is indicated by the sound of bubbles in the pump and low psi on the sand filter. For a long time I couldn't figure out why my filter wasn't maintaining pressure, then I realized I had to take the pump apart and clean the impeller, this is done by undoing the band/strap clamp between the pump and prefilter. I put a heavy rock in the prefilter basket adn that has really cut down on my need to take apart the pump to clear the impeller. Are you getting pressure in your sand filter? That might be why you're not getting filtration.

Next, try leaving your pump off for 12+ hours and see what settles to the bottom, vacuum it to waste and see how that does. I have had great success using floculant in the past, you could try it as a last resort.

For some reason my sand filter is very innefective at removing algae, best to let it settle and then vacuum to waste.