Agitator is loose, but it seems to be working. Not sure it is filling all the way. You can move it either direction with you hand.
Not sure why the agitator is loose. It is held to the shaft with a single 7/16 bolt. You can access it by removing the fabric softener dispenser and the cover that is underneath it to check the bolt.
The water level may be the result of a faulty water level switch. First blow out the tube connecting the switch to the tub and if that doesnt help try changing out the switch.
I have the agitator out now. It is the top half of the agitator that seems to have some plastic teeth that hold it in place as it agitates. That is the part that is wobbling.
Those are called agitator dogs and they make the agitator ratchet. Try replacing them and see if that fixes your problem. Bad dogs do not usually make the agitator wobble but they may in your case.