For a bacckcountry trip in Needles, you will need a permit- numbers are very limited, so if you don't already have one, you will need some luck on your side. The official park campgrounds will fill up quickly and there is no reservation system there- BUT there is open camping on BLM land just before the park entrance on 122 towards Hamburger Rock and Indian Creek Falls, as well as south of the park on 107 towards Cathedral Butte.
I would say Needles will be less of a zoo than, for example, Arches. If you do manage to score a backcountry permit in any regions of the park, you will easily be able to get away from the crowds once you are beyond normal day hiking range. Near trailheads, however, solitude will not be the norm. It all hinges on getting that permit...
If you have the inclination to look farther west, parks like Capitol Reef will be much less crowded. Other areas, however, such as the Dark Canyon or Grand Gulch Wilderness, are not nearly as rigid in their permitting and are a bit more off the beaten path. So I would say the biggest challenge in being out a week in Utah this time of year is not so much the crowds as a reliable water supply. Good luck!