It sounds like the auger always turning is the auger engagement handle/line/connections are not adjusted correctly. They/it is too tight causing the auger to engage before it should. It may be up at the handle where you loosen a jam nut, screw in or out a threaded shaft, and retighten the jam nut. (If you have ever adjusted a bicycle brake it is similar.) (I have a craftsman, made by MTD, and I think I saw MTD also list Ariens.) Or it could be an adjustment somewhere else (but I'm not sure where).
As far as one auger moving and the other not, you may have a shear-pin (shear bolt as the manual calls them) on each side of the auger's shaft (my craftsman has one on each side), and the one on the side that stops turning is sheared off. (It moves without snow present due to the driveshaft inside the outer auger shaft turning, but it won't turn under load (snow).) You may confirm a sheared-pin by turning the machine off and try to turn the auger blade. If it turns somewhat easily, that's a bad shear pin. Remove the old shear pin. If the old one is as long as the new one, that aint the problem. The shear pin in the image is one kind. Mine appear as a straight bolt (no grooves), threaded the entire length. You will have to rotate the auger blade until it aligns with the hole in the shaft. Use a nail that fits and rotate the blade until the nail goes in. Remove the nail and screw in the shear bolt:
The auger clutch control up at the handle looks like this:
From my manual
- To check the adjustment of the auger control, push forward the left hand clutch grip until the rubber bumper is compressed. There should be slack in the clutch cable.
- Release the clutch grip. The cable should be straight. Make certain you can depress the auger control grip against the left handle completely.
- If adjustment is necessary, loosen the hex jam nut and thread the cable in (for less slack) or out (for more slack). Recheck adjustment. Tighten the jam nut when correct adjustment is reached.
Here's another view of that:
I hope the above info helps you take care of both issues.