I will second the motion to "call the rangers." They often will know and share what they know about dispersed sites that you could fit into. That said, by 7pm Friday all those spots may be taken anyway.
If you were showing up mid-afternoon on a weekday, you'd be in much better shape. I'm reminded of the time I was driving south from Capitol Reef toward Boulder UT on Hwy 12, about 3 pm. It sure is a pretty drive and it made me wish I could camp up there on the mountain and enjoy the view of the valley below. Well, lo and behold, I passed a visitor center up there and I turned around and went in. The fellow behind the counter was very happy to suggest a couple of great spots that could work for my rig. I pulled off the highway about 50 yards and set up under the evergreens. A short walk put me right on the edge of a steep dropoff where I could sit and enjoy the view. You could see for miles and miles.