Why does my washing machine slam around like crazy during the spin cycle?

Nine times out of ten on a top loader that age, this is worn suspension rods. Those are the four rods that hold the whole tub assembly up from the corners of the cabinet, and they're basically springs with friction pads. When they wear out, the tub can swing way too far during spin and starts smacking the cabinet. That matches your symptoms exactly, expecially the fact that leveling the feet only helped for a couple of loads. Leveling fixes a slight wobble, but it can't fix a tub that's got too much play in it.

Easy way to check: open the lid and grab the edge of the drum. Push it side to side, then toward each corner. A healthy machine has a little give but springs right back. If it flops around like a pendulum and keeps rocking after you let go, or if it feels like it drops or clunks as it settles, the rods are done. Also peek down between the inner drum and the outer tub with a flashlight while you're in there. A bra wire or a sock wedged in that gap will thump like crazy on spin, and it's a two minute fix if you find one. If it's the rods, don't just replace one. They wear together, so you swap the whole set of four. On most brands that means opening the lid panel, unplugging the old rods from the tub corners, and hooking in the new ones. Maybe an hour of work, and the parts are cheap compared to a service call. Unplug the machine before you start pulling panels, obviously.

One more thing worth ruling out: a cracked or loose counterweight block on the outer tub will do the same thing, so while the panels are off, look for a big concrete or plastic weight and check it's tight and not crumbling. But given the age and the gradual worsening, I'd put my money on the rods.

Did this exact job on my old washer last fall and it took me way less time than I expected. Tip that saved me a headache: take a phone photo of the rods before you pull them out, because I mixed up the orientation on one and had to redo it. Also grab the model number off the sticker inside the lid before ordering, since there were like three different rod sets for machines that look identical. Mine were so worn the plastic cups on the ends had actually cracked. New set made it run quiet for the first time in years, felt like a different machine.

First time posting here so hope this is useful. My situation was a front loader instead of a top loader, but it walked across the floor on spin too and it turned out to be shot shock absorbers rather than anything to do with leveling. Same lesson though, the feet were never the real problem. What helped me was finding the parts diagram PDF for my model online, it showed exactly where everything sat so I wasn't guessing. I'd second the photo tip from the reply above, I skipped that step and regretted it. Also keep the old parts untill you're sure the new ones fixed it, just in case you need to compare.