Can I file my 1099-NEC form without filing the rest of my taxes?

No.   ALL of your income needs to be entered on the same tax return or you will have a big mess.   If you file only the self-employment income and then have to amend when you get W-2's, it will take months for the amended return to be processed.  Wait for all of your tax documents so you file one correct, complete tax return.

1099-NEC forms must be filed by those who paid money to others by January 31. Did you pay someone and need to file the form or did you get paid? If you received the form, your deadline is April 15 to file it with your taxes.