Hi Chet,
With Maytags, the two letters at the end of the serial number is the date code, in this case, "KW".
If you can turn the agitator, while it's on the splined drive shaft AND the transmission is engaged, it sounds to me like the splines within the agitator may be stripped or seriously worn! I could be wrong, but that's what I'm thinking. If that's the case it would also explain why the agitator won't stay in place when it's running with water.
Kevin