Nitrates could be an issue for more difficult corals (e.g. Acro), but not GSP and zoas. If you have high Alk {>9) then you must keep nitrates measurable or slightly higher - exactly what Russ was saying. Corals die with high Alk and no or very low nitrates. The high Alk tends to promote duller coloration but higher growth rates according to Red Sea. When there are no nitrates and/or phosphates to accompany the faster growth rates the corals starve and die.
I don't understand why you are having problems if you are using RO/DI water and a quality salt and mixing to 34-35 ppt/1.025 (at 78 deg). I would not add any other chemicals to your tank - no Alk, cal or Mg - with only having 1 or 2 corals. What salt are you using? It should be a popular name brand of some sort (reef crystals, Red Sea coral pro, tropic Marin, etc). Regular water changes should be all you need to keep these first few corals alive and happy. Only when you get a bunch of corals where you measure daily drops in Alk should you have to start adding Alk, cal & mg outside of just weekly water changes. Watch some YouTube videos from brstv regarding dosing after you get 5-6 frags looking healthy.
Next we should consider toxins of some sort. Have you ever added copper to treat fish disease in the tank? Copper is toxic to corals at medication dosage levels for fish. Any insecticides sprayed anywhere around the tank?
Outside of water chemistry, you could have flow and lighting issues. Do not blast zoas with direct flow (I.e pointing a power head directly at them). You should see them swaying very gently in the water. They will close and then die if the flow is too great. GSP is largely considered a "weed" coral to a lot of folks - including me where it is over growing some other corals. It should grow quite easily under a range of flow. I would not directly blast water at them though. Neither of these corals like super bright light. New hobbyists have a tendency to put too much light on corals. I would put any new corals down on the sand bed until they seem to be happy and then find the right placement for them long term by reading - I like the vividaquariums.com placement recommendations as a guide.
I am unfamiliar with your lights. If they are adjustable LEDs, you could try dialing them to 30-40% for now.
I would start by doing a series of 4 weekly 30% water changes with your RO/DI and quality salt. Don't add anything else to the water besides fish food. After that try another zoa or maybe a small leather or mushroom. Put them at the bottom of the tank on the sandbed and see if you can keep them happy. None of these corals like high flow or lights - look for very gentle swaying. I'll upload a video later showing you what gentle swaying looks like for GSP, zoas, and a mushroom. Keep us posted on your progress and good luck!