Should I repair or replace my 13 year old Kenmore washer?

From the age and that it is a Kenmore top load, I'm going to guess that it is a Whirlpool built machine. If that assumption is correct, I will suggest that the skate plate has worn enough to cause the "shake and rattle thing". The skate plate is the triangular piece of metal that the tub slides around on. It is hard to believe that a little bit of wear on the painted surface could cause the severe shaking, but, trust me, it is the case. The plate will cost around $ 65.00 or so and can be purchased from Whirlpool, Sears or any Appliance Parts retailer. It is not difficult to replace so if you have had no other issues with your machine, I would replace it. While you are at it, you may want to replace the drive coupler (motor to transmission) as it will probably be worn sufficiently to warrant replacement.