Retired painter's opinion.
Sand the h*ll out of it, with a sander. Start with the roughest grids and as it comes off, follow with finer grids of paper. Then lightly spackle again and sand that out by hand to smooth it as much as you can. That's probably about all you can do. Seams are really hard to smooth out.
If you're planning on a smooth finish to your new ceiling, those seams will still show through, no matter what you do. They can be camouflaged somewhat by applying another kind of texture (patterned or just random texture) to the ceiling. But you'll never hide them completely. Unless you dig/cut out all the seam paper etc, and reapply new paper and spackle, and sand the h*ll out of that.
Whatever you end up with, it is important to PRIME those areas with a primer, not just extra layers of paint, before painting again. And sand the primed areas lightly too, before painting. If you don't prime, the paint color will bleed out or "flash" (you'll see kinda shiny areas) wherever there's a seam or patch.