Drainage Issue

I live in an older home (1940) and yesterday my kitchen sink stopped draining. I have removed the P trap and, well everything under the sink, except the disposal, and they are all clear. I ran 25 feet of snake into the wall and found nothing. Now this is what my problem looks like; when I run water into the righthand side of the sink, that has the disposal, it slowley fills, as if the disposal needed to be run, so I did. Then the right side clears but the water comes up the left, when the disposal is then turned off the water levels out in both sides. If anyone out there has any ideas i would love to hear them.

You have a stoppage elsewhere in the line. Handheld snakes often are hard pressed to do much in stubborn clogs. I'd suggest either renting or purchasing a larger snake or admitting defeat and calling a plumber. I had a clog very similar recently and it required cutting in a cleanout further down the line even with a much more robust snake.

I'd suggest skipping that step and going right to the plumber!