As to gear sloppiness, I would not assume that the part you ordered matches what was the original. I've seen rubber like gears, and plastic/nylon looking gears. I don't know if it's too late, but I would consider replacing all the gears with a new set.
The gears have to line up such that there is not too much tension or pressure on the teeth. I can't remember exactly how the vertical shaft adjusts, but I remember trying to lift the gear slightly before tightening to line it up. As to the motor, I think there is a flexible part in the hand wheel, where the worm gear on the motor syncs in, and this creates some "delay" when starting up. This is important because when you are ready to reset the timing you want to cycle the machine in the proper direction to take up all this slack before tightening gears, meaning don't turn it backwards at all in order to line up the needle bar marks.