You have a Kenmore belt drive washer, a platform that is no longer
produced, but parts are widely available. There are magnetic coils
mounted to the gearcase which control various actions in the washer.
One operates agitation/spin action, and the other operates the pump.
Your problem is some aspect of the pump action. The default position
is for the pump to drain out. When the coil is energized, a lever
moves and switches the mechanical pump into a different position, in
which it doesn't pump out the water. So, what is probably hapenning
is that one of the following is broken:

Wiring to the coil that operates the pump lever, check hot side and
neutral side for breaks
The coil itself, which operates pump pump lever position
The plunger that is inside the coil
The pump

My experience shows that the most common problem is the first one.
Since default is for the pump to be in drain position, then once the
washer goes from fill into agitation, then it will start to drain at
the same time, and when the pressure switch sense lack of water, it
will start to fill again. Once you fix the reason that the pump isn't
switching to "maintain water" position, then it won't empty at the
wrong times. If the problem is the wiring, then you can either fix
the wire using your own supplies, or you can buy a nifty wire kit
(IIRC, 285610) with wires ready to be spliced. Make sure that your
connections are not near the coils, since there is a lot of movement
in this area and connectors won't hold.