Several things can cause the brine tank on your water softener to fill with water. Without a model number, I can't tell much about how old or which model of water softener you have.
Most all Sears water softeners have basically the same operation. This is what should happen as your water softener softens the water.
The water you use passes through the resin tank before it goes to the soft outlets in the house. The resin beads attract the molecules of lime and calcium and some iron to them and in effect clean the water as it passes through the tank. The resin bed has a limit to how much calcium and lime it can contain so periodically the softener goes through a regeneration cycle to clean the calcium and lime from the resin beads so they can do their job.
During the regeneration cycle the valve runs water into the salt tank for a period of time. The float valve in the brine well of the water softener limits the amount of water allowed into the brine tank. The amount of water allowed into the tank is dependent on the level the float on the brine valve is set at. The tank can fill with water if the brine valve is not stopping the water from coming into the tank.
The valve then moves to the brining cycle and sucks the water out of the tank until the brine valve stops the water flow. If this water is not being removed the tank will be as full of water as the brine valve will allow. It would require a failure of the nozzle and venturi and the brine valve to cause the tank to fill full. This is not common, but is possible.
The valve then moves into a backwash and a fast rinse to remove the salt from the resin tank. This part of the cycle should not cause the tank to fill with water.
Some other things that can cause the salt tank to fill with water are leaking o-rings on the valve that mounts on top of the resin valve or a crack in the resin tank.
If the resin tank has a crack in it the cost of repair would be expensive. The nozzle venturi and brine valve are relatively inexpensive. A resin tank and labor to replace would probably equal the price of a new water softener.
I hope this helps you decide whether to repair or replace the water softener. If you would like to repair it yourself, send another email and include the model number of the water softener and some detail of any checks you have made and how long it takes it to get full of water, and we will be glad to try to assist you.
During the regeneration