You've gotten lots of good suggestions, and I'll particularly second the suggestion of sleep headphones. I have a pair which are basically soft foam discs tucked inside a headband, and it lets me listen to white noise apps, or audiobooks, or whatever I want to, without impinging on the experience of others. I don't use them often while camping, but I do bring them when I'm in the kind of campground you're describing. It helps a lot for those occasional nights when someone is being thoughtlessly disruptive! I think you can also get under-the-pillow speakers, which might be easier to adjust to, but I'd be cautious...noise carries really far on a quiet evening in camp.
And to add my voice to the chorus, I camp a ton, and all-night electronic noise in a campground is just bad manners. Unless you're in a campground where generators are literally running all night long, I guess (which are generally not tent-friendly places in my experience, but it's been a long time since I spent the night in an eastern state park campground, so maybe I'm forgetting how different things can be). Nobody goes camping so they can hear a stranger's electronics, music, white noise, or whatever run all night long.
I'll add that I think practice helps a lot. Like your family, I sleep with a fan at home, and it's incredibly important to my sleep habits. But I also camp a lot, and my tent feels at least as much like "home" as my bed does. So have fun, wear yourselves out during the day, sleep under a light sheet if it's hot (and definitely open up the fly, or take it off altogether!), and repeat the experience as often as possible, and I bet you'll find that the camping environment gets more comfortable and sleep-inducing with practice! It's really just part of learning something new, and I'm so glad to hear that you all had a great time even with less-than-perfect sleep.