Most Sears Kenmore washers (Whirlpool based) use a threaded clutch hub which applies the spin clutch when the main motor reverses during the spin cycle. This is not an easy item to access but you can disconnect power and water, open the unit, and observe the clutch assembly on the bottom of the drum. If the clutch disc is oily or greasy it is likely to slip. This is common when a transmission shaft seal fails and a telltale sign is oil/grease drips accumulated under the washer or inside the bottom frame pan.

If this is the case, you can temporarily clean the clutch disc with a solvent (follow all precautions for use, ventilation, and fire safety) and use the machine. It will eventually require more extensive repair, including a seal, clutch disc, and possibly more parts.

There are other brands which have a lot simpler and more reliable transmission, such as the Maytag Orbital units.

It might be beneficial to check the clutch first. It could be as simple as a wire off, sticking spin brake shoes, clutch slippage in spin, or lint or foreign material preventing full clutch engagement.

If your analysis determines a major repair, you might want to weigh the cost of repair versus replacement.

I am hesitant to suggest a brand, but I have a Maytag A104 that is still running since 1986 (one of these days I suppose I should find the OFF switch...). I have another Maytag LAT8206 in the second floor laundry which is only a baby, having run only since 1998, although I did need to replace a seal on it about 4 years ago.