The large white clip on the basket drive is very loose-could that be the problem?
There all loose.
Also the clutch is a little oily, but not excessively
There should be no oil on the clutch.
The oil will make it slip. The clutch gets oil on it because the top seal on the transmission leaks.
Can you post a photo of the clutch?
Here is a photo of the inside of a washer cabinet that had a very leaky transmission and the clutch would spin the oil around onto the walls of the cabinet.
Does the inside of your washer cabinet have a line of oil on it about the same height as the clutch?
A lot of times the oil line inside the cabinet is not as dark looking as the one in the photo, but you will notice it if it's there.
In any case there should be no oil on the clutch.