Is it a slow drain or a full backup and nothing drains? When it was draining did it ever gurgle or burp as it drained?
Snake it again but this time have someone close by the gray tank and listen for the snake bottoming out in the tank. 6' should have you in the tank, but verify just to be sure. If the snake is at the tank and the line is clear as you suspect then you might have a situation with the under sink air vent and draining is creating a vacuum.
Look under the sink for one of
THESE or something similar. It's going to all the way in the back of the cabinet up behind the sink. Just follow the drain line to the Tee and go up. It should be just slipped into the drain pipe and not glued so you can remove it and test the drain. There should be one for all the drains. Vanity, shower, clothes washer and such. Anyplace there's a trap will have the vent close by. You won't find many of these in older domestic plumbing systems due to the multiple roof vents that were used in the past. RV's they are pretty common with the actual roof vent is to vent the tank and not the individual interior drains.
Good luck.