Thank you for your question and I understand your concern.
First, the agitate dogs will not cause or prevent the basket from spinning. The agitate dogs are used to make the agitator ratchet in one direction whenever there is enough friction applied to the augur. To remove the agitator you will need to remove the fabric dispenser and the screw/bolt that you see down the center of the agitator. Once the bolt has been removed, grasp the base of the agitator and jerk it up and it should come off.
You did not make it clear if the washer motor is running during the spin cycle or not. If it drains the water out and then you open and close the lid and it does not begin spinning, then it may have a broken brake release actuator or a worn clutch.
I added a YouTube video I found with
on how to access and replace the clutch band. You can order the part from