No dead algae does not dissolve out. Typically it turns white or brown once from green algae once it is dead, and has to be filtered out over time, or vaccumed to waste.
Your first objective is to obtain a proper test kit such as the TF-100. or Taylor K-2006 which is lacking in CYA and FAS-DPD reagents with the TF-100 being the better option. Pool store test results are inconsistent and deemed unreliable.
Your second objective is to do a little light reading on Pool School here in the forum, get familiar with the terms and procedure TFP utilizes, and purchase a case or two of bleach or pool chlorine. No shock packets, or Tri chlor pucks or tabs. What shocking a pool really is, elevating your level of free chlorine or FC to heightened and sustainable levels appropriate to your CYA ratio. Shock is not necessarily a product you purchase in granular packets.
Pool School - Chlorine / CYA Chart
You want to use the Pool Math Calculator to determine how many fl.oz. of what strength bleach to add to take your pool to shock levels, and ensure you are killing off and filtering out dead algae. Brushing is important as well as algae has a biofilm that is slimy and the chlorine needs to penetrate it to kill it off.
SLAM and maintain your shock levels everyday until your OCLT Overnight Chlorine Test loses no more than 1-2ppm FC in 24 hrs. Your CC or combined chlorine is less than 0.5 ppm and your water is clear. If you allow the SLAM to fall or fail before meeting these 3 criteria, you risk the probability of algae redeveloping and starting over.
As I mentioned earlier, skimmer socks or hairnets over the skimmer basket will help you in filtering out the dead algae, but you must remain vigilant on the SLAM to kill it. Sand filters can take awhile so you have to be patient. If you still have difficulties getting the water crystal clear before it back and I have another solution.