Windshield wipers don't shut off and washers don't work

Try replacing wiper motor relay and check wiper circuit wires for short or damage

Could be the wiper/washer timer relay is faulty if the car is fitted with one. Modern cars tend to have a single timer ECU that does everything that needs timing. Older cars used a relay to control the wiper single sweep timing and the number of continuous sweeps after using the washers.

Another possibility with the wipers only could be the internal switching in the motor gearbox.

The two problems aren't necessarily connected. Try testing the washer pump with an independent supply and then with an independent ground, then try both...

So that wipers park properly, the wipers have 2 power sources. One is from the dash switch to turn the wipers no and off, but the second is the parking power, which is always supposed to be on. The parking power is supposed to be shut off by a rotating cam inside of the wiper motor housing itself. If you don't want to take it out and fix it, you could always just pull the parking wire to the wipers, and then try to manually park them by shutting off the wipers at exactly the right time.