I'm not completely sure how the drive elements connect up on this. I know there is no belt. Gear box seems very unlikely. You should examine how the agitator attaches to the drive shaft below the top cap of the agitator. Sounds like it's slipping, and if that's the case, it may not be able to spin either. If it works differently than I think, it could be the clutch slipping. To get to the clutch is major surgery. You'd have to remove the agitator and would discover any slipping parts here on your way to the clutch any way.
VICTORY!
Bottom line: clean the discharge pump filter first if you get this code. Mine was clogged with years of accumulated junk (coins, pencil parts, keys, lint, etc.).
What I did: cleaned filter, cleaned all contacts (both ends) on CCU, MCU, interlock switches with Dentek Easy Brush, standard size, dipped in rubbing alcohol. Tested door switch, interlock switches and motor: all good.
To access the filter, remove three screws holding bottom front panel and remove panel. The discharge pump filter is located at the bottom right. Twist the 3" round white cover counterclockwise to remove the filter. Put a pan below to catch water. Note: I also suggest you temporarily disconnect the black hose coming down to the back of the filter from the washer and shake it to dislodge any junk in the hose.
Update 7/1/14: F-28 returned. Resolved by disconnecting the CCU motor connector MS2 and pinching in on the prongs to tighten the connection. Used a sharp pointed pick, and pressed in on both sides until they touched.
Update: 8/25/14: Got F-28 again. Disassembled the MI3 electrical connector (three blue wires)on top row of the CCU. This is the connector that was pinched as described above (Error: It was the MI3 connector instead of the MS2). The tiny metal clips inside were deformed from the previous pinching. Reformed them and reassembled. Problem solved for now.
Update: 10/16/14: The saga continues. Got F-28 again. Soldered MI3 wires directly to board on CCU and added bullet connectors to connect wires. Then, a couple of days later, got code again. This time removed the MCU (from front, after removing bottom panel and filter assembly). Observed that board contacts for the MI3 wires were oxidized. Took tiny jewelers flat-head screwdriver and scraped off oxidation. Also sanded wire-end connectors with folded 220 grit. Then compressed wire-end connectors slightly to make for tighter grip. No problem in a month. Think this fix should last a few years.
Best Answer
I fix my own stuff, but don't know about your washer. However, a quick search revealed this to be a common problem & I ran across an old blog that matched your symptoms perfectly. Kenmore washer won't spin out all water from clothes.
Although, the parts involved in resolving the multiple problems, appear to total more than $300 for the parts alone. You can call a local Appliance Repairer for their estimate for the typical course of action for this common problem. But, based on just parts cost I'd suggest getting a new washer with a new warranty.