It only took another hour to book my flight :(
I found out that when you redeem United miles on the computer, they only give you SOME of the available flights on the screen. They do not give the option of going on to another screen, so the flights they offer are the "best possible matches" and not all the matches.
After speaking with the customer service rep from phone booking, I discovered that the 8 p.m. flight out of Utah did exist and I wasn't imagining things. However, he would have to charge me a phone processing fee if I booked it through him.
No way to book it online, more money to book it with him... "call the online customer service rep," he said. Of course, that rep can do it for free but you cannot. What?!?!
So back on the phone, with the other United Customer Service line. Are you as confused as I am? It makes sense, but seems really silly.
1 hour down and now the other, apparantely cheaper rep, has booked my online flights without a phone charge.
The other kicker: In order to get home in time for work on Monday, I have to fly from Utah to LAX at 8 p.m. Sunday and arrive back in D.C. at 6 a.m. Eastern Time. Then, go straight to work. Yikes!