TG, They are still wobbly, but I've deduced a few things:
1. The stalk comes out of the ground and runs along for a few inches before rising, so they aren't quite as wobbly as I initially thought. If I trace it back to where the roots actually start, it's pretty firm. They still don't seem quite right, but the problem is less severe. I think. Or I could be completely wrong.
2. You all got me thinking more about the roots, and I have been watering them with a hose on a trickle, rather than a sprinkler. That allows the water to run deeper, and force the roots to chase it. A side benefit is that I use less water and avoid leaf fungus issues. Until I get a chance to set up a drip irrigation system, it's going to be a pain to keep moving the hose, but nothing I can't deal with in the meantime.
Now, I'm also dealing with a major caterpillar issue. I plan to solve that with a solution of vegetable oil, chile peppers, and garlic liquified in a blender, sprayed onto the leaves.