I have a Kenmore washer purchased Mar. 2001. The final spin cycle is very noisy (sounds like -but isn't- an unbalanced load) what should I check first?

Probably leaking oil onto the dampers, oil from the transmission. The dampers are very touchy. Water probably going into tranny through seal just under the agitator cap (physically just under, mechanically buried), forcing oil out of tranny. 9 years of laundry, not a bad record. I'd run it till she doesn't run at all, probably another 5 years, just guessing.