The white stuff at the bottom of the pool is probably calcium carbonate that precipitated so yes, you can vacuum that to waste. You are right that some of that settled down out of the water. Your water should start to clear up since your water is closer to being in balance now IF the pool store's numbers are correct. You do not need to raise your TA at all and in fact should probably keep it on the low side to avoid the cloudiness and can keep your pH lower as well.

With the pool store numbers of a pH of 7.8, TA of 127, CYA of 10, and CH of 357, I get a saturation index of 0.3, not 0.5, so I don't know how they are doing their calculations. With the later numbers of pH 7.6, TA 99, CYA 10, CH 334, I get a saturation index of 0.2, not 0.1.

If you want to add some acid to get the pH down around 7.2, that should speed up getting rid of the cloudiness (see below about using Smart Sticks instead), but it's not necessary if you're willing to wait. Lowering the pH will help reduce the TA a little bit as well -- especially if you aerate.

Get your chlorine back up which you can do easily using the Burn Out 35 or the Smart Sticks or both (not mixed together, but the sticks in the skimmer and the Burn Out 35 in the pool). Since your CYA is low (again, if we believe your pool store), then using a chlorine source that provides CYA, such as Smart Sticks, is fine to use (note that it will lower pH so you can use those instead of adding acid as I mentioned above).

Richard