Does closing my cover-letter with salutations or closing phrases like sincerely yours or yours truly make it look like an email? If so, how best to close the letter?
Sincerely, seems like it would be appropriate just about anywhere.
Sincerely yours, or
Yours truly definitely do not sound appropriate. You would use those in a personal communication to someone that you're close with/to.
OK, I started writing letters before email was a thing. "Sincerely" and "Very truly yours" were appropriate ways to end a business letter.
If they make it look like email to you, that's because you have been taught that emails should look like business letters.
Go for it.
Assuming you are writing a full cover letter ie a direct pitch as to why you meet their requirements.
The formal way in English is:
"Yours sincerely" when you know the person you are addressing, "Yours faithfully" when you are starting your letter with Dear Sir/Madam, etc
If you mean just a five to ten line email with a word or pdf attachment - you could use the above or Just "Best Regards"