Why do my walls get paper thin at the bottom when I pull up?

Thin bottoms like that almost always come from the first pull, not the third one. Here's what's probly happening: when you start a pull at the base, you're squeezing hardest right where the clay is thinnest and most vulnerable, then easing off as your hand travels up. Your brain thinks you're applying even pressure, but the bottom gets three solid squeezes across three pulls while the top barely gets touched once. The fix is counterintuitive. Start each pull at the very bottom, yes, but keep the pressure feather-light down there and gradually load up more squeeze as your hand climbs. Basically your heaviest compression should happen at the top of the pull, not the bottom. Also check that your inside fingers and outside hand are actually stacked at the same height. If the inside finger sits a hair lower than your outside hand, it's shaving clay off the inner wall near the base without you noticing. That's a super common culprit. Another thing to try: instead of three full pulls from base to rim, do fewer passes and leave the bottom third alone after the first pull. Once a section is at thickness, just don't touch it again. Every extra pass over the base is another chance to thin it out.

As for the wobble creeping back in by the third pull, that's usally uneven hand pressure or an unanchored arm. Lock your elbow against your hip or your forearm against the splash pan so your hands move as one rigid unit. Wobbly clay also gets worse if you're drowning it in water, since the base gets soggy and starts to smear. Small dribbles only, and sponge the puddle out of the bottom.

And honestly, if two tore off at the base, they were probably past saving anyway. A cylinder that thin at the foot was going to crack in the drying no matter what. Cut your losses with the needle tool, squish it back into a ball, and reclaim it. Your 25 pounds will last way longer than you think once the pressure thing clicks.

I have found that when the rim starts looking ragged or inconsistent, it usually indicates that the clay's plastic properties are failing, which is a different kind of structural issue than simple pressure imbalance. Rather than focusing solely on the pulling mechanics, perhaps you should meticulously inspect the hydration of the clay before you even initiate the pull, ensuring it possesses enough 'slip' quality to maintain its integrity during the rotation. When a piece starts to exhibit that wobbly behavior during a subsequent passage, it is often due to the material simply losing its ability to hold a coherent wall thickness under the necessary compression. This slightly moist, cohesive state is absolutely critical for any form to acheeve the desired structural rigidity.

The pull is fine, kid. But you gotta look at the clay before it even hits the wheel. If you're starting with a messy lump that hasn't been properly wedged and smoothed out, you're asking for trouble. I spent years watching pieces collapse because the clay itself was just weak and unprepared. You need a solid, uniform mass before you try to shape it into a cylinder. If the consistency is off right at the center, the wobble is going to come back no matter how gentle your hand is. What kind of clay body are you using? That might be the real issue.