Lava Point or Try the Back Country
Lava Point is a First come First Serve campground. It is small so get there early. You will take Kolob Terrace Road North. This will get you away from most of the crowds near Springdale. My suggestion is get a back country pass and take you pack and tent on one of the many side trails off of Kolob Terrace Road. You will see some beautiful country and you won't have to fight the crowds for a campsite. You are also at a higher elevation so it is a little cooler than down in the main park.
This is one of the most beautiful places on earth!