I've got a Kenmore that was built in 1994, so it may or may not be the same as yours. I have a cap that pries off the agitator. You can just barely see a line, but it's there. When the cap is off, then looking down in, there is a nut/washer combination that unscrews.
With the nut off, I can just pull up on the agitator and it will come right out. I think the first time it stuck a bit, but not too badly. You may need to call the place that repairs or sells them. Ask them for help. They can tell you how to remove the agitator. Good luck.