I took a bit of time yesterday to head outside and do some experimenting with the solar scope to see if I could eliminate the Newton's Rings. Long story short, I couldn't. Brian at KW Telescope suggested I try a few lenses to try to break up the interference pattern, but nothing worked. I tried a moon filter, and the couple of filters that came with the camera (clear and IR?). Looks like I'll have to buy a new camera if I want to get rid of them.
That said, I was able to pull some nice images through the high clouds. Probably my best images of the Sun yet.