Alk and Ca are sort of on either ends of a balance scale. In typical tank conditions, when one goes up the other goes down and if one component goes down, the other goes up. This is why the Alk rose when Ca dosing was stopped. You didn't say how high the Ca level was at the time, but it might help for you to know that Calcium can go as high as 600 ppm with no bad effect, if the Alk is right. As you may have figured out, the SPS burning was caused by Alk being too high.

Step one is actually quite easy. Rapidly decrease the Alk from it's 13 dKH level by simply adding a lot of Ca. Hobbyists will generally not see this as an alternative, because they have not tried it, because they do not understand it, because it may seem counter-intuitive, but it works and is easier than doing water changes. This Ca addition should cause a snowing effect in the tank. Snow is a good thing. It's a chemical response to balancing the Alk and Ca levels in the salt water. Where the two components, Alk and Ca are higher (more saturated) than they need to be in the water they combine and drop out of the water (precipitate) as CaCO3(Calcium Carbonate). CaCO3 is the mineral that rock and sand are composed of. In some tanks this precipitation appears immediately as "snow" and in other tanks it may coat pumps and heaters with a layer of CaCO3 cement.

As an illustration, I help a hobbyist with a tank that is always testing on the low end of Alk (~7 dKH) and Ca(~350 ppm). Yet he has an amazing Blue Wall Hammer Coral that grows like crazy, currently filling almost a quarter of the 180 gal tank. There is also a basketball sized Candy Cane Coral and lots of other stony coral, mainly LPS but also SPS. He manually doses Alk and Ca every other day and has been advised by LFS (and by me) that he could increase the dosages. He gets nervous about this because his tank had a bad experience with dosing too much. 

What I take away from this example is that the important thing is to dose what the tank is eating but not to worry about artificially elevating levels to reach some kind of target.

Why would this help

Because the problem right now is too much of the Alkalinity component in the water. Decreasing salinity decreases Alkalinity. Alkalinity is carbonate ions floating around in the water waiting to combine with Calcium. Removing some salt water and replacing it with pure water will dilute the amount of those ions, making the water more bearable for the aquarium inhabitants.

But won't decreasing the salinity hurt the coral

Actually no, not if it's done over the course of an hour or so. In the wild, these animals are genetically tuned to having salinity go down quickly, like when a torrential rain adds millions of gallons of fresh water to the reef and tide pools. Ask anyone that has seen their top off water do a small flood of their aquarium. Salinity can drop fairly rapidly to 1.022 or even 1.020 without any bad effect on the animals. But when raising the salinity back up, do it more gradually because there is no genetic variable to handle a quick rise in salt content.

This means that if you have chosen to do some water changes, do them with the aim being to bring tank water salinity down by several points. 

Oh, one other thing. Above I said that Mg should be tested and if below 1200, should be raised. Without Mg available to the animals, they cannot consume Alk and Ca to build their home. Mg is critical to the action of the Alkalinity ions combining with Ca to form Calcium Carbonate. 

Hardly any of us realized, when we got in to this hobby, that chemistry would be so important. 

Aloha,

Mark