I have 5 66s, 2 99s, a Sewmore 404, and all of them have movement to the bobbin carrier. The bobbin carrier just lays on top of the oscillating hook. It is not fastened down. So every time the hook oscillates the carrier will move a bit.
Remove the slide plate and lift up on the little lever that extends over the slide plate spring. Lift it and rotate it forward, towards you as your sitting in the sewing position. You'll have to rotate the hand wheel and kind of feel when everything is in the right place, but when it is you can slightly rotate and lift the bobbin carrier out.
Nothing to unscrew, leave your screwdriver in the drawer.
Clean EVERYTHING. Even under the lip of the hook, all the way around. Might remove the needle plate too and clean under that.
Then lightly oil - things in there are metal to metal, needs a bit of lube. If the little wick is still in the carrier latch, oil that.
They get noisy when they are dirty or gummed up. The dirt and gook and lint tend to grab the carrier and pull it with the hook rather than let it sit freely in it's place to do it's work.
Here's a link to a place that has a ton of info with photos on just how to clean, adjust and fix your 66:
http://www.tfsr.org/publications/tec...machine_manual
Now that I have paper, I'm going to finish printing every thing on that site.
Joe