Thanks for the help, everyone. The bike shop had a Tange Levin, and the price was right, so I put it on tonight. The steering is smooth as can be, and there's no wiggling anywhere.

When I took the old one off, I confirmed that someone had created a franken-headset. On the upper assembly, the pieces simply weren't made to go together. I'm glad to be rid of it and have the right one on.