My gargage disposal is also clogged. When I turn it on the water runs to the other side. If I run water on both sides without turning the disposal on, then I just have free standing water on both sides that drains at a snails pace. I have removed and cleaned the pipes and trap (even replaced worn pipes), snaked the pipes and stack. Even though the sinks are filling up quickly, I assume the problem can't be that far down. I did snake the stack in my basement. And again, still the same problem. Other than calling a plumber, what am I not doing? Or could do?

Let see if I have this straight. You removed the trap and snaked the line inside the wall for at least 6 feet. Correct? That didn't get it so you went to the basement and located the kitchen stack and snaked from the cleanout. Still no joy, Am I correct so far?
One of two things, Either you snaked from the wrong stack or you didn't put out enough snake to reach blockage.
Now a few questions. When you snaked from the trap how much snake did you put out?
This basement stack. What size is it? Is it directly under the kitchen? How much snake did you run out? What size is your snake? How long? If you ran a hose in the clean out could you fill it up? Get back to me with the answers and we'' look for a solution. Good luck, Tom