Moire is generally a consequence of cameras without an optical low pass filter. This is a device that smoothens out the pixels which are smaller then the imagers. Both aliasing and Moire come from the same issue.
Edit: Unfortunately, I don't know if any video cameras in your price range that ship with OLPF's. However, there are some companies making aftermarket plugin adaptors for some cameras, which help reduce moire and aliasing. http://www.mosaicengineering.com/
Moire is directly related to image sharpness. You can attain "softness" through glass if you so prefer. I rarely see Moire issues on my Blackmagic Pocket camera's, mainly because I don't use high end glass, so the image is already "softer" then it would be with a higher end piece of glass. Really modern/crisp glass can exaggerate moire and aliasing issues on pretty much any camera.
I believe there are screw-on filters which help these issues, though I'm not sure how well they function. Now that you know the problem, you can do some research.