The problems you are having with tonearm position are all caused by the two thin metal levers or cam followers which move into position to stop the tonearm at different places depending on the desired result.
These are the two kind of curved pieces right near the program cam and you can see them move during a cycle. They are on the same pivot as the tonearm drive plate assembly which is the piece with the peg on it that follows the large groove in the big red main cam. There is a spring attached to the end of each one that pulls them back into rest position when the pins on the cam are not acting on them.
The grease on the pivot of these gets gummed up from age too and they must be removed, the old grease cleaned off, and then they must be relubed. I use automotive wheel bearing grease.
When these followers can't be pulled back by the springs, the tonearm will not always (or never) go to or remain at the position desired.
Sometimes you can get away with spraying a penetrating oil in there to free them up but, eventually, they will get stuck again. After you clean and relube them, move them by hand to make sure the springs pull them back to rest position every time.
EDIT: Oh and BTW, there are service and owner manuals for the 960 available in the library here, along with some other documents, so you can see what is supposed to happen in different operating modes.