A clean pipe that cause water back up to both sink.

[Catelizabeth]

At A on this picture, I noticed the little protrusion. That gave me hope there was something there that would unscrew. I think you are saying no, the thing at A will not unscrew.

There looks like enough place to put a trap adapter at B, but it would be at an awkward place.

Did you do your snaking by unscrewing C and inserting the snake?

Yes, you are correct. Only the C can be unscrewed and let the snake get in. A &B are all glued firmly.