My K2 will not move unloading auger unless engine is at half throttle or more. You might also look to see if "wrap around" chain is complete. I did have a chain come apart some years ago.
Also check to see if your pivot collar is stuck. Sometimes the ball bearings in there get jammed, especially if the machine had been left outside. Sloan Express sells a replacement collar with sliding nylon pads in place of the ball bearings.
My auger sticks too after sitting from one season to the next. The remote for your auger sticks, you need to take a tool and move the remote pin in and out and spray it with penetrating oil. This is the pin that is on your pump controls.