There's a metal cable that connects the climate control unit to whatever it is that directs the air flow. The cable could be seen underneath the dash directly (on the driver's side, near the center vents), and pulls on an air redirection mechanism (blend door? or something like it) in the same way a brake cable does on a bike. For whatever reason, the mechanism didn't have its entire range of motion, so when the setting was on "blow through dash", it still blew through the floor vents. I could physically move the mechanism though to obtain the "blow through dash" setting.
I added a small kink in the cable to help the mechanism go all the way to the "blow through dash" setting. This probably means when I turn the dial all the way the other way, it will still be blowing on the floor a bit while also blowing at the windshield, but it seemed like a good trade.