First, about the Midnight thing. I tried that for another park recently and was told to wait until 10am (I think Eastern Time).
The Watchman campground has electricity which is important in the summer and fall for to run your A/C. If you're in the self-serve campground inside the park that time of year should be OK as others have advised.
If you stay in Springdale, just outside the park the shuttle system can be caught there too. Since it looks like you have a TT and truck, you can stay as far away as you'd like (e.g. Zion River Resort is pricey and very nice), but being inside the park at dawn/dusk is pretty special. I live 45-minutes away and make regular day trips. Last November was our first camping trip inside since 2000 and we were very glad to do it.