I trust you are aware that the eastern entrance to Zion cannot be used by the larger trailers and motorhomes due to a tunnel. You can get small trailers through, but only by shutting down oncoming traffic and driving down the center of the road. This, of course, costs extra, and as well makes you popular with the tourists.
If you are doing both, I, too, would recommend Kodachrome State Park, BUT it is on the eastern side of Bryce, and not a good choice for visiting Zion.
There are three campgrounds in/near Springdale, two NP, the other private, but you would be best served by getting reservations, as they are popular. There are also a half dozen or more along routes 9 and 89 between Zion and Bryce.
