We have a Kenmore washer that is about 3 y/o and now will wash BUT then half way into rinse cycle it stops and the letters, "FL" flash and nothing we have tried will make the machine resume or do any other task.

FL may mean FLood level(or over fill), found someone with same problem: Kenmore Elite Washer Over Fill Switch? [Archive] - DIYbanter

The things that cause overfill are the water valve or the pressure switch. Many times you can blow through the hose on pressure switch and all is fine. If both are good it may just be false signal or bad faulty board