My wife and I decided to paint a little-used 10X12 bedroom in our house.
The walls had not been painted in 20 years, and the ceiling had not been
painted in maybe 30 years. On the walls, we used two coats of paint. They
look nice. The ceiling, however, is another story.

The ceiling is a common textured surface; it was white and we kept it white,
using Dutch Boy white flat ceiling paint. After the first coat, it was
obvious a second coat would be needed, so we applied a second coat
yesterday. Well, the second coat didn't help much and, if anything, it
looks worse. The job is basically blotchy, with some areas lighter than
other areas, even though we used over a half-gallon of paint on the recoat.
So, it needs to be done again and any advice would be appreciated.

Since it hadn't been painted in 30 years, should we have primed it first?
I've never primed previously painted walls in the past. Or, is it possible
the Dutch Boy isn't good paint? Or, should I use a better roller cover? (I
used one with a 1/2" nap that said it was for textured ceilings, among other
things.) What should be done next to finish this job properly? Thanks.