The reason I asked about your CH level is because if the calcium levels are too high, then they can cause cloudy water, especially if you're using cal-hypo to chlorinate. You're correct in that you don't really need a minimum calcium level with a vinyl pool, but calcium levels that are too high can still be a problem for you.

Does the pressure gauge on your filter work? Is it rising as you're stirring up the stuff on the bottom? When is the last time you backwashed the pool?

My 3 co-mods also have sand filters, and have used a small amount of DE added through the skimmer (just enough to cause your pressure filter to rise by 1 psi) to help coat the sand and make it filter smaller particles. You could try that, in case the particles causing your cloudy water are just too small for the sand filter to get. If your pool is that cloudy, and your filter pressure isn't rising, then it means that your filter isn't able to catch the particles, for whatever reason. The DE m ight help.