Go very slowly with the sanding, 2000 grit, also sand wet if you can and keep wiping dry to check when you have the surface smooth. you only need dampness not pots of water, and you can manage the water on the surface so it doesn't get where you don't want it to be.
Don't sand the edges at all, they don't need it, if you are patient you won't need to as you will get close.
Keep the sandpaper clean, make sure you don't have any grit between the paper and the surface. After you have the surface matte and smooth, then use cutting compound to bring up a shine.
You can hear surfaces scraping when you have something under the sanding surfaces if you listen carefully
Again, make sure the cloth is super clean with no sandpaper bits in it or you will end up with swirls