Spoiler alert: It worked on the first try, calling 1-800-421-4655.

Update 11/1/19: Just saved another $75 for a last minute United award I decided to book. I called 1-800-4655 number, and the first agent I spoke to tried to charge me close-in booking fee for moving my flight from December 2 to November 10. I hung up and called back, same number, and the second agent did not mention it whatsoever and re-booked for free.

United MileagePlus charges $75 to book an award less than 21 days before departure.