Thank you, Kevin.S and garylearntech, for your input.
I've been playing with Hitfilm Express for about a week now (Pretty cool stuff!) and it seems to be making things better, though there are many artifacts popping up.
I gave Super Sampling a try, making sure Anti-Aliasing was selected. I think it's as smooth as ever, if not more, which is kind of the problem. Fast moving things naturally have some sort of blur, like when you wave your hand in front of your face quickly. It's just kind of...ow to watch what I have without.
My current idea is to try to create a "green screen" within CTA for separate parts. Basically, throw in a green background and export separate videos of the different moving bits of the animation. Then hopefully, I can apply motion blur individually and just stack everything together as one video, all without the extra mess. I am thinking just applying it to the full screen is just not good and may be why what's happening is happening.
If anyone has any extra input or any different ideas, I am all ears! I think Hitfilm and the Super Sampling will prove useful, so I appreciate the input so far very much.