I assume that you are on a mixed UA/Star ticket. Use the ANA availability tool to see if award space opens up on the flights that you want. I find it much easier to obsessively/compulsively check using this tool than to repeatedly call. As soon as I see space, I call UA Rez to change my ticket.

If the tickets were booked using miles from the 1K's account, you can make changes with no fee. If the miles are from your account, there is a $100 fee every time you switch.