Hi Duke,
What you have is a 1965 model Lady K set. They were very popular and very likely in production well into 1966.
I don't know what to tell you about the loud squeal, except I would suspect the basket drive pulley. Are you still having that problem? If so, does it go away when the machine goes into spin?
As to the transmission that does not properly engage, I have heard of this a few times. Try pushing down on the agitator and see if that firmly makes it engage. If it does, then the one or both (probably both) of the springs that push down on the drive gear have either rusted and broken or fatigued.
On a healthy belt-drive transmission, engagement of agitate is accompanied by a loud slapping sound, which is made by the drive gear dropping down onto the agitator shaft hubs. These are two large, pinky-finger diameter hubs that protrude from the agitator shaft by about 1/2 inch. There is a strong spring above the drive gear in between it and the cover of the transmission, and another slightly smaller gear around the wigwag post. Both gears in concert act to put a great deal of downward force on the drive gear, and when the gaps in the drive gear meet the hubs in the agitator shaft, they come together rather forcefully and agitation is engaged and will stay that way until the wigwag cam bar activation causes the drive gear to be pulled back up (still under load of the springs) and become separated from the shaft hubs.
When the springs begin to go bad, the engagement becomes less forceful, and eventually the moving drive gear will slip over the top of the hubs and cause a knocking or minimal agitation. If the transmission is smooth and quiet once it finally engages, I myself would not hestitate to take it apart and replace the springs. However I did see one once that had chewed up shaft hubs from so much slippage and the shaft had to be replaced (now NLA). IT IS NOT EASY TO REINSTALL THE AGITATOR CAM BAR WHEN NEW SPRINGS ARE INSTALLED. By far the easiest thing to do is replace the transmission for another one if you can locate one.
Gordon