"new tank syndrome" usually refers to a tank without the proper colonies of nitrifying bacteria that grow during the "cycle". Maybe some other people use it to talk about acclimating fish, I have always heard it used mainly by new hobbyists when talking about the cycle. But that's just semantics.
As for what to do, at this point your tank is not properly cycled, and there are a few ways to go about it. If you do nothing and let it run by itself, the cycle will take place over about 4-6 weeks. Maybe longer since you only have the one little fish who won't produce much ammonia. This can be very tough on the fish because ammonia will have to spike, as well as nitrite, before colonies of bacteria which eat them form. It is possible to have fish live through an unassisted fish-in cycle, but also very likely the fish don't survive. If you have access to anybody with a well established freshwater tank, you could take some of their filter media and use it to seed your tank. Some good fish stores will sell cultured bacteria pads as well from their filters. This is usually the method I use, as there will already be a good amount of the bacteria tranferred, and the ammonia and nitrite won't have to rise. Another way is to go with bottled products. Personally I have used Dr. tim's one and only many times and I think it works quite well. Seachem stability is used by some hobbyists I respect as well, so probably good stuff. It can be used in conjunction with seachem prime to keep ammonia in a form that the bacteria can eat, but won't harm the fish. These products will shorten the cycle usually to about a week or two. During the cycling period don't change the water, as you will disrupt the cycle.
Another note- why are you raising specific gravity, and what are you using to do this? African cichlids can live in brackish water but do not need to, and it adds another parameter that can get out of whack. Evaporation can cause jumps in salinity and at 1.012 your water is pretty salty already. personally I would keep the cichlids in fresh water to have one less thing to worry about, unless you plan on having a dedicated brackish setup with fish that require it.