Just after posting, I think I figured it out. The tiny screw that locks down the gear apparently was the problem. The metal portion of the feed tube is very close to where the screw rotates as the stepper spins the gear. It was just close enough that it was catching the tube and hanging up. It was basically getting stuck and could not feed the filament. It was able to make the faster rotation because there was more pressure and it forced the gear to fight through. The slower continuous feed was not enough pressure the make the stepper fight through.
Basically all I did was move the gear further onto the shaft of the stepper. There's still enough "teeth" to feed the filament, but it's close to edge. As long as it works, that's all that I care. I hooked the lever back up so I could feed filament in, and it's running like it was before the upgrade now. So check the tiny screw and make sure it isn't snagging the metal portion of the filament tube, or any other part that's causing the gear to hang up.