After you file

Yes, if your original 2023 return was e-filed within TurboTax, you should be able to e-file the amended return as well.  Please see this link for more information.

As for potentially mailing the Form 1040X and updated 1040, you only need to sign Form 1040X as this form replaces your originally filed Form 1040.  Please see the instructions here as well as the excerpt below to help you determine if you need to attach any other documents to your amended return if you need to mail it in.