I actually had to do exactly this last summer when my Wi-Fi cut out right as I was trying to snag a deal. I was worried about a long wait, but honestly, it was pretty painless. The best way to handle a phone booking is to have your flight numbers and credit card ready before you dial.
The agent I spoke with was super helpful and was even able to modify my seating preference on the spot. Just a heads-up: they sometimes charge a small ticketing fee for phone bookings, but for the peace of mind, it was totally worth it for me!
Would you like me to find the direct customer service number for United to save you a search?