Washer doesn't agitate

My washer will not agitate, and in the final spin, while spinning it lets water in the tub. When it’s done there is about an inch of water in the tub and I have to wring the clothes out by hand to put in the dryer. I had a repairman out and he said the timer needed replaced. He replaced it but still the same problem. Then he said the board on the timer needed replaced and he did that. Still the same problem. Now he says the selector switch needs replaced, but doesn’t know if that will fix the problem. I need help. My clothes are pileing up. Will that fix the problem or is it something else wrong? Please help.

Lots of times a failing timer will damage the motor. If you can use an amp probe, measure the motor current and post results.
You also should replace the repairman unless he can give you more measurement data. The motor speed selector switch should be checked per the schematic.