Welcome to Just Answer.
Sorry to hear about your problem.
Since the disposal is working fine, it sounds like there is a partial clog in the drain line,past the disposal. You can try using a plunger or a plumbers snake/ auger down the sink drain.
Second option is to remove the trap and use a product called the "Drain King," found at your local home improvement store or hardware store; they work great at unclogging drains. It comes in different sizes for different drain diameters and you simply attach it to a hose and slide the hose down the drain pipe or at removed trap, as far as it will go. You then turn on the hose and it builds up pressure to the point where it releases in several bursts, expanding and contracting. You can clear one obstruction and then push the hose further to get to others and they are only around 10 to 15 dollars.
Here is a short video on how the Drain King works, but ignore the foreign language.
Chemicals often do more harm than good on the plumbing, so stay away from them.
If you need further help or clarification on this answer, please do not hesitate to reply to this post.
Thanks for asking your question at Just Answer.
Best of Luck, Brian
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There is a possibility that the clog is directly below the disposal, before the trap. With the trap off try running the water with a bucket under the disposal. This may release the clog with a little help from an auger or metal clothes hanger.