So, after I replaced the washer shock springs, it made it through a handful of loads before going out of balance and making lots of noise again. The basket does not move out of place within the tub, so it is not the bushing. It seems I did not get good enough rods from amazon.
I finally got a good look inside the washer while it was running, and it was definitely the suspension rods not damping the motion of the wash tub again.
After a bunch more research I came across a video on how the suspension rods work and how to refurbish them, https://m.youtube.com/watch?v=IZZCwXEmGhE, and one of the comments said that replacing the foam in the rod with butcher’s twine held up for years. One of my roommates had a bundle of cotton twine that I could use to attempt the repair.
A rod that came out of the washer, it still has lots of high friction grease on it.
After I got it opened it up, the foam insert was nowhere to found.
So, I wrapped a wad of string, just big enough to fill the leftover space in the pink cup, around the rod, getting it well covered in the traction grease, and pushed it all together.
After doing all four rods, a spin test made it through the full range of speeds.
It also made it through an unbalanced load of laundry with no interventions or rebalancing, I hope it keeps working through the next loads.