I'm from Moline in the Quad Cities and grew up camping on the Rock River in Joslin. That campground is out of your way, but...
I camped at Scott County Park with my kids several times and loved it. But, that was 25+ years ago. From the photos, it appears very nice still. I tent camped so it was quiet and private, but the main campground was pretty full. It is a popular campground. Big swimming pool and a Pioneer Village next to the park. It's located just north of Davenport in Eldridge (close to I-80).
https://www.scottcountyiowa.gov/conservation/scott-county-park
Also camped fairly often at Loud Thunder Forest Preserve in Andalusia, Illinois. If you look at a map of the Quad Cities, you can see that I-280 goes around the QC to the west. I-80 goes around the QC to the east. So if you take I-280 it will put you pretty close to Andalusia and Loud Thunder. Loud Thunder shouldn't be crowded at all.
https://www.ricfpd.org/loud-thunder-forest-preserve
Then there are the Army Corps of Engineers campgrounds outside Iowa City in Coralville. All look very well kept.
https://www.recreation.gov/camping/gateways/158BTW, the Maquoketa Caves and campground are very popular, but that is 30 miles north of Scott County Park in Eldridge, IA (see above).
https://www.iowadnr.gov/Places-to-Go/State-Parks/Iowa-State-Parks/Maquoketa-Caves-State-Park
Edit: Forgot about Illiniwek Forest Preserve. It is right off of I-80 just before you cross the Mississippi River into Iowa. My brother camped there often and I stopped by his campsite once. IIRC, it seemed crowded. I never camped there, but went sledding there as a kid. Maybe check it out, since it is right off I-80.
https://www.ricfpd.org/illiniwek-forest-preserve
I think Army Corps of Engineers has campgrounds in/or near the Quad Cities, too. They weren't there when I was growing up, but I'm pretty sure I saw them on the USACE website.