Don't use drywall in this instance. It is too heavy for your application, and attaining a clean "Oval" shape is not easy with that material anyway. As far as liquid nails, I would bet that the existing ceiling has at least paint if not paint and texture on it, so the liquid nails will only hold as well as the paint its spread on (guess what...)
Screwing into the existing ceiling is questionable because it will be hard to locate the joists or trusses above the existing surface.
If you are still with me, and you choose foam, go with a non-colored high density foam.
I don't have any better information than this, your project is a little out of my experience. Others please feel free to critique this advice and provide your own.