Hi Justin, welcome to aw.org. I also agree with previous posts saying the most likely culprit is the water pump being frozen up by old water scale.
There is a three part service manual to these Kenmore Machines available on Automatic Ephemera if you wish.
Kenmore Washer Service Manual Part 1
Kenmore Washer Service Manual Part 2
Kenmore Washer Service Manual Part 3
This is a 1957 Model, here is the Fall/Winter '57 Sears Catalog page with your washer, it was sold for $144.95 (which is $1282.74 in today's dollars)...

There is a three part service manual to these Kenmore Machines available on Automatic Ephemera if you wish.
Kenmore Washer Service Manual Part 1
Kenmore Washer Service Manual Part 2
Kenmore Washer Service Manual Part 3
This is a 1957 Model, here is the Fall/Winter '57 Sears Catalog page with your washer, it was sold for $144.95 (which is $1282.74 in today's dollars)...
