Rear end sways over bumps?

Also, I feel like a shudder in the steering wheel at all times, not just at high speed.

It sounds like you could have a tire that is coming apart internally. I would raise the rear of the car and spin the tires and watch the tread, if you see an irregular pattern the tire needs to be replaced.

This guide can help you raise the rear of the car:

https://www.2carpros.com/articles/jack-up-and-lift-your-car-safely

If the tires are okay look at the swing arm mounting bushings these can fall apart allowing the rear end to have unwanted movement. I have included a diagram below of what I am talking about (#6). Check out the images (below). Let us know what happens and please upload pictures or videos of the problem.

Just had tires replaced because one had a bulge on the sidewall. I replaced all 4. And the bushings look fine. I checked those when I changed the shocks.

Could it possibly be because I need an alignment? Like am I supposed to get an alignment after changing tires or shocks?

Yes, that is possible. A video of the problem would help us see what's going on as well.