Do I need to sign and mail in my tax forms to IRS and State or is my electronic tax submission is done?

There is nothing to mail unless you were told to do so when you e-filed your tax return.

See this - https://ttlc.intuit.com/community/print-and-file/help/do-i-need-to-mail-anything-after-i-ve-e-filed/01/26144

Unless you have been specifically told to submit a form on paper you don't need to mail anything if you have filed electronically.