My hat is off to Peter Tung. His observation about the nasty water dripping from the vanes inside the drum was the clue I was looking for. Rather than try to clean them by pouring cleansers/bleach into them, however, I removed them and got a nasty surprise.
The vanes are very easy to remove. There is a single, phillips screw in a recessed hole near the back of the drum. Unscrew it, and then gently slide the vane toward you. It will unlatch and you can lift it up. Each one was covered with the same slimy crud.
Once you get all three out, take them to the sink and remove the white piece that holds the black balls in place. Be careful not to break the little tabs holding the white piece in. I found that just a little pressure on the black balls will pop it out. Make sure the little balls can't fall down the drain before you start.
I took them apart and scrubbed the pieces with hot soapy water (you'll need an old toothbrush for the small crevices).
While I was doing that, I ran the machine on "sanitary" with HE washing machine cleaner; bleach and detergent would probably work, too. After that, I wiped inside to remove any residual gunk, and re-installed the vanes.
Only time will tell, but I'm pretty sure this will solve our problem.
PS - I had previously cleaned the seal (which was also full of slimy mold) and that didn't fix the problem. Its certainly good to do that for other reasons, but it won't solve this problem by itself.
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No liquid detergent
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