I have no experience on that particular model of Arts-Way grinder but I know on the bigger newer models there is a snap ring on the end of each shaft to hold the gear tight.     You set the backlash of the gears by putting shims behind and in front of the gears.   

And there is a square key in each shaft/gear for power transmission.


I hate to be the bearer of bad news but that looks bad and expensive unless you can handle it yourself, machine shop $'s will quickly exceed the value of the mill.      As someone else mentioned that feed looks wet/sticky and the grinder waaay too rusty for trying to push that through the auger.

At the very least you need to run the auger fast (pto speed) and just dribble the feed into it.    Slow and full = lots of torque.   Too much torque finds the weak spots like you did.