It look like dried up water. Like when you have a water stain left on something. Ive seen this before after doing a water repair and leak stopped. I would rip it all down from ceiling to wall and re-sheetrock it. That is after all leaks have been repaired.

I had a painting business and when I came across areas that needed a lot of spackle id call a drywaller, Thats all they do and they do it well. Worth the money.