Linda,
First you'll need to clean all the old gummed up oil out of the mechanicals. Then properly oil it. The sluggish movement of the cam follower is the best clue you gave.
Then, well this might should be first, get an owners manual. All the knobs must be set in the correct places for the reverse or the pattern settings to work. As good as Kenmores are, and I do like them a lot, they are not user intuitive. I have to use the manual to get mine to work correctly.
If you have the stitch length set on a wide stitch, the reverse lever will move farther than if the stitch length is set on a small stitch. And if you do not have the controls in the right place, the stretch stitches with require reverse mechanism movement to work, will not work.
Joe