Here's a online free manual you can download and print for a Tanaka post hole digger, as your title stated that's what you have.
However, I noticed that you wrote you have a Badger Plus auger, so I tried to find info on that, but a manual doesn's exist outside the manufacturer, you can call them to see if they'll mail you one.
I's suggest you look at the gearbox, as it sounds as if you've sheared a key inside the gearbox on the main drive shaft, based on your description. The gear box is straightforward with a simple 12:1 gear reduction ratio and uses only two (2) gear wheels and shafts.
I couldn't attach the picture but here's a link that will show the gear box, just scroll down once it loads. It should be easy to open and repair the gearbox, as there isn't much to it. Drain the gear oil and separate the halves and you can check for a sheared key on either of the gear wheels and replace the broken one. They're designed to shear to prevent damaging the more expensive gear wheels, etc.
Hope you find this Very Helpful and best regards!