I accidentally claimed a dependent that should of been claimed by my spouse. How do I correct both returns?

Did you already file it? Was it accepted?

Is there a reason you are not filing a Joint return?   Joint is usually the best way to file even if you are separated.

Changing a dependent after your income tax return has already been filed and accepted is not a complicated process, however, it will take some effort on your part.

• First, you must wait for your original return to be processed and your refund to be issued to you. 
• Second, the person who should have claimed the dependent must file their income tax return claiming the dependent by mail.  They cannot E-file their return.  It will be rejected.  They can mail in their return at any time. 
• Third.  You will have to file an amended return to remove the dependent that should not have been claimed. 
• Lastly, removing a dependent will cause you to lose credits such as Earned Income Credit, Child Tax credit, and others.  So be prepared to pay back some of the refund you received. 

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Yes. I'm waiting for the 1040X form to be available. In the meantime, I just received my tax return in my account.

The 2020 Form 1040X does not become available until the end of March 2021 (subject to change).

https://www.irs.gov/forms-pubs/new-information-for-form-1040-x-filers