Flowing Spring was an interesting area. The only bathroom was very close to the main highway (less than a mile) but the road was VERY wash-boarded and had a drop off to one side. The scenery on the way in is more high desert than lush green forest. The bathroom area has a large parking lot that could easily accommodate a lot of cars but all the spaces to camp there will be walk in only. The trees aren't super dense so expect a decent amount of sunlight.
We traveled a bit further down the road and found a spot right over the bridge and next to the river. This site (as well as a lot of the others we saw) was not very level. There was a lot of powder fine dust at the camp site and every time the wind kicked up it was a giant dust cloud. There were trees around but not a lot of shade. I'd say that had we gone when the weather was perfect to be outside in the sunlight and there was no wind blowing the dust the site would have been really nice.
Towards the end of the road there are a few more places that you could park a small trailer to camp. We arrived on a Thursday and almost everything we saw that would accommodate a trailer already had people in it. I'd guess we saw around half a dozen places you could put a small trailer (the website says nothing over 20 feet). There were plenty of places that will accept tents especially if you're willing to walk in.
We walked the river for a ways and the water was really dirty in the pools and you couldn't see much deeper than a few inches unless it was running. We didn't see any evidence of trout in the river but honestly didn't try fishing either. A lot of the pools didn't have enough water to connect them so I assume the river was just very low when we went.