No agitation or spin, but water pumps out.

If the washer will fill and drain, but won't agitate or spin, you, indeed, may have a broken motor coupling. The noise of the motor engaging, lack of agitation and ability to still drain are the usual symptoms. This is a common fail item with this model washer through normal wear and tear. Newer versions of the coupler are more resilient and resist breaking. The following link explains how to replace one:

http://www.fixya.com/support/r385250-replacing_motor_coupling_kenmore_top

The motor coupling is located between the drive motor and the transmission and takes the place of a drive belt. They do wear out and can break. The replacement part number is included in the link I provided and is an improved upgraded coupling that is more resilient. It should last longer. If you have questions, or require additional assistance, please let me know. I hope this helps you.

A replacement coupling can be purchased on line at appliancepartspros.com for about $11. Just use the part number in the link and "Search by Part Number" for your search criteria. Let me know if you have any questions. I hope you find this information helpful. NOTE: There are other problems that can cause the washer symptoms as you describe, but the motor coupling is the least expensive and easiest solution to try first. If the motor coupling is still intact, please let me know.