Jack
I'm having the same exact problem. I think it might be easy to fix - at least I was told it was. The good news is that you don't have to take the machine apart from the back.
Take the top round lid off the agitator and remove the thing with the handle on it - I think this is the place where you pour the fabric softener. I don't use it though. If you do use it for fabric softener it will probably look very black and ominous - don't worry about that, it's just old softener.
If you clean it up you'll see 4 plastic things that are called "directional cogs" or "dog ears" - that might be what has to be replaced.
I haven't repaired mine yet - lack of tools and I'm not into mechanical things.
You could go to this page on this site: http://www.partselect.com/MultiModels.aspx?ModelNum=LA9100X&SearchTerm=cogs&RefineSearch=0
It's the result of a search I used for the Whirlpool LA9100X (my model #) - you can put in your own if it's not the same.
But the diagram may give you an idea of what you'll be looking at/for and the site has pictures of parts that you can probably get from your local appliance store if you have one.
A friend of mine had this done and that's the only reason I knew about this - he didn't fix it though - has some kind of insurance for appliance repair but he said the repair guy did it in a few minutes and told him it was a common thing.
Hope this made some sense to you. Good luck. And should you attempt to repair it yourself, let me know how do do it. :))
jack, January 2009
