How would you clean the weep holes of a garbage disposal if your sink only drains when the garbage disposal is running?

A disposal creates centrifugal force and acts as a pump. Often a blocked or partially blocked sink drain can result in slow drains and a disposal will force water through. If it really is the disposal, try slowly adding coffee grounds and/or fibrous material like broccoli centers or pineApple centers while cold water and the disposal are running. If that doesn't solve the problem, have the drain cleaned and make certain that the drain vent works.