Drain cycle.
The way this washer's mechanical systems work is as follows:
There is one motor, it can run forwards, stop, and run backwards.
During fill, the motor is off.
During agitation, the motor runs forward. This runs the transmission via one belt and the pump (running backwards) via the other.
During drain, spin, and spray rinse, the motor runs in reverse (driving the pump forward).
When the washer starts to spin/drain, the motor runs backward at full speed (or slow speed if "gentle spin" is selected). This runs the pump forward at the same speed as the motor, pumping water out. At the same time, the transmission belt intentionally slips in lieu of a clutch. This is why the pump can run at full speed while the tub starts to spin slowly and then accelerate faster and faster to its maximum spin speed of 618 RPM.
Assuming both belts are intact and connected, the only way the pump can run in the direction to pump out water is when the tub is spinning.
The sound you likely hear when you turn on the washer is the water valve solenoid engaging. I would disconnect the hoses from the spigots and test the spigots to see if they are getting water pressure and flow (have a bucket handy). If they both work, check both ends of the washer hoses. They often have screens which can become clogged with mineral deposits, pipe debris, or 35 years of silt and dirt in the water. If the hoses get good flow and pressure (the hoses have the same threads as a garden hose so you can test them outside), then check the screens on the washer's water valve where the hoses screw on. The screens should be visible as soon as you remove the hot and cold fill hoses from the back of the machine.
If ALL of this checks out OK, then please let us know.
As for the timer, if the timer advances on its own, then the motor is OK. If it fails to stop, it sounds like a chipped cam which would necessitate a new timer. (It could be a sticking contact which would require opening the timer and cleaning or a new timer). A timer replacement isn't a bad job, you just have to take the wires off the old one and put them in exactly the same places on the new one. Make a drawing and take pictures before you start. New timers can be ordered through most appliance parts stores or they can be found on ebay. If you do get a new timer, save the old timer motor! Those motors are getting harder to find and aren't getting any cheaper.
The hoses are usually fine. If you can grab them and squeeze them to see if they are still soft, they are fine. If they are rock-hard, they could stand a replacement. If you get a leak during fill from the back left corner of the washer, it's just the injector which costs under $20.00 and takes about 10 minutes to replace. If you have mineral deposits in your water valve, I would replace the water injector anyway.
Yes, it's worth fixing,
Dave