You will love Omaha's world-class zoo. Some people think it's the best zoo in the country, even if it doesn't have pandas.
As for local campgrounds with full hookups, only the West Omaha KOA has those facilities, as far as I know. However, both the zoo and this KOA are right off of I80 and a 30 minute drive apart.
If you can survive a few days with electric only, I would suggest Walnut Creek Lake in Papillion, NE, an unusually clean and well kept city campground with concrete back-ins and large grassy sites. There are no showers available, but the cost is only $14 a night.
You might also want to consider Lake Manawa State Park in Council Bluffs, IA, which might be the 2nd closest campground to the zoo. I believe the fee is $16. The closest campground is the Bluffs Run Casino & RV Park at $30, and I believe they do have electric and water. This is a parking lot affair with great views of the interstate, but only 5 minutes from the zoo.
I hope this helps.
Rig: '97 30' Limited
Towed by: '13 Silverado 2500 Duramax