No, you do not have to continue to file a Schedule C, or SE tax for the other income reported on your Form 1099-MISC. Your other income will be reported on Form 1040 Schedule 1 line 21.
To enter your 1099-MISC with TurboTax open in wages & income select 1099-MISC follow the program prompts to enter your information. TurboTax will automatically transfer the data to the proper tax forms