Thank you for the update on the leaking of the water softener.
You may have a leak in the resin tank. I would suggest removing all the salt and all the water and then pull the power plug on the softener. Then watch the salt tank for a couple of days to see if you get more water in the salt storage area.
If that proves to be the problem with the tank leaking, and because of the softeners age I would suggest replacing the softener, because of the cost of replacing the resin tank. You can use the model number which should start with 625.3XXXX and put it into Sears Parts direct to get a price on the resin tank for this model.
If it does not leak at the resin tank then the float valve could be the problem and may need to be removed and cleaned as illustrated in the troubleshooting web site.
Please let me know what you discover.