Take the trap off the drain line(s), put a bucket under the open down tube(s), turn on the faucet and see if water flows freely. If it does then the problem is further down the line. Many small snakes (augers) will poke a hole in the clog without actually removing it. When the snake is removed the "hole" clogs up again. Either get a stronger snake or call a plumber/drain cleaner with the proper equipment to do the job correctly.
Also, concerning a plunger. Does the dishwasher connect to the disposal? If so, use some locking pliers to "pinch" off the connecting hose, along with closing off one side of the sink. Without pinching off the connector, your allowing water to be forced into the hose towards the air gap and NOT the drain pipe.