Adding your handwritten signature is optional, but it's recommended for more formal cover letters. (uptowork.com)
The rule is that the more authority the position has then the more formal the cover letter should be and the more specific. (exclusive-executive-resumes.com)
In a cover email, it's not necessary (after all, it'd look awkward to have random personal contact information at the top of a post), but with formal cover letters, it becomes trickier. (skillcrush.com)