Your description does sound like a clutch problem, if not physically bad than at least oily from the gearcase.

Quick but potentially dangerous test: when empty put it in spin and depress the lid switch so you can open the lid without interrupting the spin. Start it and let it spin up a little and with a heavy glove on see how easily you can stop the basket by pressing on the side of the moving basket. Once it speeds up it should not be easy to stop it.

Safer test: If you have gotten the cabinet off, look at the clutch, it's the round shiny silver thing above the gearcase, it should be spinning faster than the basket initially but basket speed should catch up. If the clutch spins at high speed and the basket isn't that's your answer.

The clutch and gearcase spin at motor speed and the clutch slips to accelerate the basket without forcing the motor to bring the basket up to speed instantly. Usually a new clutch is the solution but you can possibly clean the pads and lining to save $.

I'd just replace the clutch if it was me. They are not very $ and you going to have to take it all the way out to clean it anyway. But it's worth watching a video to see if you want to handle the repair yourself. Look for Whirlpool/Kenmore Direct Drive clutch replacement videos.

Your part and a video: Whirlpool 285785 Clutch Assembly - AppliancePartsPros.com

There is also the possibility of the basket drive tube needing action if the plastic drive cam has worn, but it is less likely. This is a more involved repair, requiring a special spanner wrench to remove the basket.