Your drain is plugged downstream from the disposal. The disposal isn’t the cause of, or the soluton, the problem. It’s an easy fix, but I am not sure I can explain it to you.
In the cabinet underneath the sink, there’s a pipe (hopefully PVC) that leads from the bottom of the disposal that takes the broken down food and water to the sewer (septic field). It will have a collar that tightens and seals the pipe to the disposal exit. You need to put a bucket below that pipe, then unscrew the collar which will release the pipe, and all the scraps and water that are in the sinks/dishwasher will come gushing out (hence the bucket.) If you’re lucky, and in my experience you will be about 90% of the time, the plug will be in that section or just past it, and a screwdriver or a coat hanger will be enough to break up the clog. If so, reattach the pipe and run water through the drain for a minute or two. Check for leaks under the sink.
Most of the time this will solve the problem, but if not it is time to call a plumber. A handyman with a drain snake will be cheaper, it really isn’t something that requires a plumber in most cases.