Turn off continuous rasterization of the pre-comps (the little sun symbol). Distortion effects operate on pixels, using CT re-rasterizes the layer over and over again to retain resolution, so distortion effects cannot work.
However, now that I have turned off continuous rasterization the comp quality is terrible. Is there a way to have both effects & cont. rasterization when I render out to a .mov or can I only have one or the other?
Is there a way to have both
No. If, as you say it looks "terrible", then it simply has insufficient resolution and then no algorithm on this planet can "invent" (interpolate) pixels in a way that still would make it look sharp. Your only option is to rebuild everything at higher resolution then.