My whirlpool washing machine model LA9500T suddenly today stopped advancing from cycle to cycle. It fills water correctly and goes into agitator cycle. It then keeps agitating until I manually advance to next cycle, on and on. What can I do to fix this and is it repairable by me (I can do small stuff). Is it going to be expensive in parts? This is an old washer, about 16-17 years old.

Thanks for help.

My first guess according the way you describe the problem is the timer motor went bad. On the older timers you could buy a timer motor separately and being that old it's a possibility. Call an appliance parts store with the model number and they should be able to tell you if this is the case.
Timer motor may run around $40 and the complete timer anywhere from $70. To $120.