Well, I sort of figured it out. I'm actually suprised I hadn't alreasdy tried this one, but by turning off the DEPTH channel in the Brush attribute, it fixed it. Now I can see all of my rings. They overlap each other now and don't look so good, but I can tweak it and make it better. I'm just happy they all show up.

I wonder if there's a way to set the depth distances.