I will again ask, when you set it for a spin cycle, what happens? Is there water in it now? Does the motor make any noise when a cycle is selected? Have you checked the lid switch as it will prevent the motor from running (and pumping water) during the spin cycle. You said the tub overflows when you start a cycle. Is that with water already in the tub? Does it agitate?

I ask questions and I need answers in order to provide some assistance.

If nothing happens when you set it for spin, then I am still leaning toward the lid switch. You can check and make sure it clicks when you open close the lid, but the only real way to check would be to do a resistance check using a multimeter. The second problem of overflowing water is still the pressure switch as it is what tells the timer to start agitating and shuts off the water at the desired level. Both could be checked with a multimeter and unfortunately, they are not easily accessible without removing some panels. That is the advice I have for you so if you decide to do it yourself, or get someone else to look at it, we are here to help.

Not knowing what part of the world you are in, my suggestion would be to visit some place like Appliance Repair | Company Reviews & Ratings, Appliance Repair Articles, see some of the technicians in your area and see how they rate with their customers.

If the sounds is coming from the motor, it sounds like it is either locked up, or unable to start and is tripping the overload thermostat on the motor. This is usually accompanied by a hot electrical smell. Several things can lock up a motor from the pump to the gear case or even a failed capacitor preventing the motor start winding from starting the motor. You can remove the motor from it's mount and set it on the ground with the wiring still connected and see if it will run then. If it does, then you must be dealing with a mechanical problem, but if it doesn't, then an electrical problem is what you are looking for. The motor not running would explain why it won't drain, but not why it keeps filling with water. A bad timer could cause both problems if the noise seems to come from the timer. Sounds like the problem you are having is not as cut and dried as many others we see. Complex or multiple failures can be tough to locate and that is where being able to see something first hand allows for a much better diagnosis by a professional. That would be my suggestion unless you want to keep looking.